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❓Let a=4^{0.9}, b=8^{0.48}, and c=( frac {1}{2})^{-1.5}. Then ( )A: c>a>bB: b>a>cC: a>b>cD: a>c>b
💡Okay, so I have this problem here where I need to compare three different expressions: a, b, and c. They are given as exponents with different bases and exponents. Let me write them down to see clearly:a = 4^{0.9}b = 8^{0.48}c = (1/2)^{-1.5}Hmm, the question is asking me to compare these three and determine their order from largest to smallest. The options are A: c > a > b, B: b > a > c, C: a > b > c, D: a > c > b.Alright, so I need to figure out which one is the largest, which is in the middle, and which is the smallest. Since all of these are exponential expressions, maybe I can express them with the same base so it's easier to compare.Let me think about the bases: 4, 8, and 1/2. I know that 4 is 2 squared, 8 is 2 cubed, and 1/2 is 2 to the power of -1. So, maybe I can rewrite each of these expressions with base 2. That should make it easier to compare their exponents.Starting with a: 4^{0.9}. Since 4 is 2^2, I can rewrite this as (2^2)^{0.9}. When you raise a power to another power, you multiply the exponents. So, that becomes 2^{2 * 0.9} = 2^{1.8}. Okay, so a is 2^{1.8}.Next, b: 8^{0.48}. Similarly, 8 is 2^3, so this becomes (2^3)^{0.48}. Multiply the exponents: 3 * 0.48. Let me calculate that. 3 * 0.48 is 1.44. So, b is 2^{1.44}.Now, c: (1/2)^{-1.5}. Hmm, 1/2 is 2^{-1}, so this is (2^{-1})^{-1.5}. Again, multiply the exponents: -1 * -1.5 = 1.5. So, c is 2^{1.5}.Alright, so now I have all three expressions in terms of base 2:a = 2^{1.8}b = 2^{1.44}c = 2^{1.5}Since all these are powers of 2, and since 2 is a positive number greater than 1, the function y = 2^x is an increasing function. That means the larger the exponent, the larger the value. So, to compare a, b, and c, I just need to compare their exponents.Let me list the exponents:a: 1.8c: 1.5b: 1.44So, 1.8 is the largest, followed by 1.5, and then 1.44. Therefore, a > c > b.Looking back at the options, that's option D: a > c > b.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For a: 4^{0.9} = (2^2)^{0.9} = 2^{1.8}. That seems right.For b: 8^{0.48} = (2^3)^{0.48} = 2^{1.44}. Yep, that's correct.For c: (1/2)^{-1.5} = (2^{-1})^{-1.5} = 2^{1.5}. That also looks correct.So, exponents are 1.8, 1.5, and 1.44. So, a is the largest, then c, then b. So, the order is a > c > b, which is option D.I think that's solid. I don't see any mistakes in my reasoning. It's always good to express everything with the same base when comparing exponents because it simplifies the comparison a lot. Since 2^x is an increasing function, the larger exponent means the larger number. So, I'm confident that D is the correct answer.**Final Answer**boxed{D}
❓If p ^ 2 – 13p + 40 = s, and p is a prime number between 1 and 20, inclusive, what is the probability that s is a multiple of 5 and s < 0?
💡First, I need to identify all prime numbers between 1 and 20. These are 2, 3, 5, 7, 11, 13, 17, and 19.Next, I'll substitute each of these prime numbers into the equation ( s = p^2 - 13p + 40 ) to calculate the corresponding values of ( s ).For each prime number:- ( p = 2 ): ( s = 2^2 - 13 times 2 + 40 = 4 - 26 + 40 = 18 )- ( p = 3 ): ( s = 3^2 - 13 times 3 + 40 = 9 - 39 + 40 = 10 )- ( p = 5 ): ( s = 5^2 - 13 times 5 + 40 = 25 - 65 + 40 = 0 )- ( p = 7 ): ( s = 7^2 - 13 times 7 + 40 = 49 - 91 + 40 = -2 )- ( p = 11 ): ( s = 11^2 - 13 times 11 + 40 = 121 - 143 + 40 = 18 )- ( p = 13 ): ( s = 13^2 - 13 times 13 + 40 = 169 - 169 + 40 = 40 )- ( p = 17 ): ( s = 17^2 - 13 times 17 + 40 = 289 - 221 + 40 = 108 )- ( p = 19 ): ( s = 19^2 - 13 times 19 + 40 = 361 - 247 + 40 = 154 )Now, I'll check which values of ( s ) are both multiples of 5 and less than 0:- ( s = 18 ): Not a multiple of 5 and not less than 0.- ( s = 10 ): A multiple of 5 but not less than 0.- ( s = 0 ): A multiple of 5 but not less than 0.- ( s = -2 ): Not a multiple of 5 and less than 0.- ( s = 18 ): Not a multiple of 5 and not less than 0.- ( s = 40 ): A multiple of 5 but not less than 0.- ( s = 108 ): Not a multiple of 5 and not less than 0.- ( s = 154 ): Not a multiple of 5 and not less than 0.None of the calculated values of ( s ) satisfy both conditions of being a multiple of 5 and less than 0. Therefore, the probability is 0.
❓the length of a rectangular floor is more than its breadth by 200 % . if rs . 300 is required to paint the floor at the rate of rs . 5 per sq m , then what would be the length of the floor ?
💡First, let's denote the breadth of the floor as ( b ) meters.The problem states that the length is more than the breadth by 200%. This means the length is 200% of the breadth plus the breadth itself. Therefore, the length ( l ) can be expressed as:[l = b + 200% times b = b + 2b = 3b]Next, we calculate the area of the floor. The area ( A ) of a rectangle is given by the product of its length and breadth:[A = l times b = 3b times b = 3b^2]We are given that Rs. 300 is required to paint the floor at a rate of Rs. 5 per square meter. This means the total area can also be calculated by dividing the total cost by the rate per square meter:[A = frac{text{Total Cost}}{text{Rate per sq m}} = frac{300}{5} = 60 text{ sq m}]Now, we equate the two expressions for the area:[3b^2 = 60]Solving for ( b ):[b^2 = frac{60}{3} = 20][b = sqrt{20} = 2sqrt{5}]Finally, we find the length using the relationship ( l = 3b ):[l = 3 times 2sqrt{5} = 6sqrt{5}]Approximating ( sqrt{5} approx 2.236 ):[l approx 6 times 2.236 = 13.416 text{ meters}]
❓A curve is described parametrically by[(x,y) = (3 cos t - 2 sin t, 5 sin t).] The graph of the curve can be expressed in the form[ax^2 + bxy + cy^2 = 9.] Find the ordered triple ((a, b, c)).
💡Okay, so I have this parametric curve given by the equations (x = 3 cos t - 2 sin t) and (y = 5 sin t). The problem is asking me to express the graph of this curve in the form (ax^2 + bxy + cy^2 = 9) and find the ordered triple ((a, b, c)). Hmm, alright, let's break this down step by step.First, I remember that when dealing with parametric equations, sometimes it's helpful to eliminate the parameter (t) to get a Cartesian equation. In this case, the parameter is (t), and I need to find a relationship between (x) and (y) without (t).Looking at the parametric equations:(x = 3 cos t - 2 sin t)(y = 5 sin t)I notice that both (x) and (y) involve (sin t) and (cos t), so maybe I can express (sin t) and (cos t) in terms of (x) and (y), and then use the Pythagorean identity (sin^2 t + cos^2 t = 1).Let me try to solve for (sin t) and (cos t). From the second equation, (y = 5 sin t), so I can solve for (sin t):(sin t = frac{y}{5})Okay, that's straightforward. Now, let's plug this into the first equation to solve for (cos t):(x = 3 cos t - 2 sin t)Substitute (sin t = frac{y}{5}):(x = 3 cos t - 2 left( frac{y}{5} right))Simplify:(x = 3 cos t - frac{2y}{5})Now, let's solve for (cos t):(3 cos t = x + frac{2y}{5})(cos t = frac{x}{3} + frac{2y}{15})Alright, so now I have expressions for both (sin t) and (cos t) in terms of (x) and (y):(sin t = frac{y}{5})(cos t = frac{x}{3} + frac{2y}{15})Now, since (sin^2 t + cos^2 t = 1), let's substitute these expressions into that identity:(left( frac{y}{5} right)^2 + left( frac{x}{3} + frac{2y}{15} right)^2 = 1)Let me compute each term step by step.First, compute (left( frac{y}{5} right)^2):(left( frac{y}{5} right)^2 = frac{y^2}{25})Next, compute (left( frac{x}{3} + frac{2y}{15} right)^2). Let's expand this:(left( frac{x}{3} + frac{2y}{15} right)^2 = left( frac{x}{3} right)^2 + 2 cdot frac{x}{3} cdot frac{2y}{15} + left( frac{2y}{15} right)^2)Compute each part:(left( frac{x}{3} right)^2 = frac{x^2}{9})(2 cdot frac{x}{3} cdot frac{2y}{15} = 2 cdot frac{2xy}{45} = frac{4xy}{45})(left( frac{2y}{15} right)^2 = frac{4y^2}{225})So, putting it all together:(left( frac{x}{3} + frac{2y}{15} right)^2 = frac{x^2}{9} + frac{4xy}{45} + frac{4y^2}{225})Now, substitute back into the Pythagorean identity:(frac{y^2}{25} + frac{x^2}{9} + frac{4xy}{45} + frac{4y^2}{225} = 1)Let me combine like terms. First, let's handle the (y^2) terms:(frac{y^2}{25} + frac{4y^2}{225})To combine these, I need a common denominator. The denominators are 25 and 225, so 225 is the common denominator.Convert (frac{y^2}{25}) to 225 denominator:(frac{y^2}{25} = frac{9y^2}{225})So, adding the two:(frac{9y^2}{225} + frac{4y^2}{225} = frac{13y^2}{225})Now, the equation becomes:(frac{x^2}{9} + frac{4xy}{45} + frac{13y^2}{225} = 1)But the problem asks for the equation in the form (ax^2 + bxy + cy^2 = 9). Right now, the equation equals 1, so I need to multiply both sides by 225 to eliminate the denominators and make the right-hand side equal to 225. Wait, but the desired form has 9 on the right. Hmm, perhaps I should adjust accordingly.Wait, let me think. If I multiply both sides by 225, I get:225 * ((frac{x^2}{9} + frac{4xy}{45} + frac{13y^2}{225})) = 225 * 1Which simplifies to:25x^2 + 20xy + 13y^2 = 225But the problem wants the equation equal to 9, not 225. So, perhaps I can divide the entire equation by 25 to make the right-hand side 9.Wait, 225 divided by 25 is 9. Let me try that.Divide each term by 25:(25x^2)/25 + (20xy)/25 + (13y^2)/25 = 225/25Simplify:x^2 + (4xy)/5 + (13y^2)/25 = 9Hmm, that gives me:(x^2 + frac{4}{5}xy + frac{13}{25}y^2 = 9)But the question asks for the equation in the form (ax^2 + bxy + cy^2 = 9). So, comparing, I can see that:a = 1b = 4/5c = 13/25Wait, but let me double-check my calculations because I might have made a mistake when combining the terms or scaling.Wait, let's go back a step.After expanding, I had:(frac{x^2}{9} + frac{4xy}{45} + frac{13y^2}{225} = 1)To get rid of the denominators, I can multiply both sides by 225, which is the least common multiple of 9, 45, and 225.Multiplying each term:225*(x^2/9) = 25x^2225*(4xy/45) = 20xy225*(13y^2/225) = 13y^2So, 25x^2 + 20xy + 13y^2 = 225Now, to make the right-hand side 9, I can divide both sides by 25:(25x^2)/25 + (20xy)/25 + (13y^2)/25 = 225/25Simplify:x^2 + (4xy)/5 + (13y^2)/25 = 9So, that gives me the equation in the desired form:(x^2 + frac{4}{5}xy + frac{13}{25}y^2 = 9)Therefore, the coefficients are:a = 1b = 4/5c = 13/25Wait, but let me check if this is correct by plugging back into the original parametric equations.Alternatively, maybe I made a mistake earlier when combining the terms. Let me re-examine the step where I combined the y^2 terms.Original equation after substitution:(frac{y^2}{25} + frac{x^2}{9} + frac{4xy}{45} + frac{4y^2}{225} = 1)Combining y^2 terms:(frac{y^2}{25} + frac{4y^2}{225})Convert to a common denominator:(frac{9y^2}{225} + frac{4y^2}{225} = frac{13y^2}{225})That seems correct.Then, the equation is:(frac{x^2}{9} + frac{4xy}{45} + frac{13y^2}{225} = 1)Multiplying both sides by 225:25x^2 + 20xy + 13y^2 = 225Dividing both sides by 25:x^2 + (4/5)xy + (13/25)y^2 = 9Yes, that seems correct.Wait, but in the problem statement, the equation is given as (ax^2 + bxy + cy^2 = 9). So, comparing:a = 1b = 4/5c = 13/25So, the ordered triple is (1, 4/5, 13/25). But let me check if this is correct by plugging in the parametric equations into this Cartesian equation.Let me take a specific value of t, say t = 0.At t = 0:x = 3 cos(0) - 2 sin(0) = 3*1 - 0 = 3y = 5 sin(0) = 0Plug into the Cartesian equation:a*(3)^2 + b*(3)*(0) + c*(0)^2 = 9Which simplifies to 9a = 9, so a = 1. That matches.Now, let's try t = π/2.At t = π/2:x = 3 cos(π/2) - 2 sin(π/2) = 0 - 2*1 = -2y = 5 sin(π/2) = 5*1 = 5Plug into the Cartesian equation:a*(-2)^2 + b*(-2)*(5) + c*(5)^2 = 9Which is 4a -10b +25c = 9We know a =1, so:4*1 -10b +25c =9Which simplifies to:4 -10b +25c =9So, -10b +25c =5Divide both sides by 5:-2b +5c =1Now, from our earlier result, b=4/5 and c=13/25.Let's plug these in:-2*(4/5) +5*(13/25) = -8/5 + 13/5 = ( -8 +13 )/5 =5/5=1Which matches the equation. So, that's correct.Let me try another value, say t = π/6.At t = π/6:cos(π/6) = √3/2 ≈0.866sin(π/6)=1/2=0.5So,x=3*(√3/2) -2*(1/2)= (3√3)/2 -1≈ (2.598) -1≈1.598y=5*(1/2)=2.5Now, plug into the Cartesian equation:a*(1.598)^2 + b*(1.598)*(2.5) + c*(2.5)^2 =9Compute each term:a≈1*(2.554)=2.554b≈(4/5)*(1.598*2.5)= (0.8)*(3.995)=≈3.196c≈(13/25)*(6.25)= (0.52)*(6.25)=≈3.25Adding them up: 2.554 +3.196 +3.25≈9.000Which is approximately 9, so that checks out.Therefore, the coefficients are a=1, b=4/5, c=13/25.Wait, but in the initial problem statement, the user had a different approach and got a different answer. Let me check their steps to see if I missed something.Wait, no, in the initial problem, the user had:They expanded the expression and set coefficients equal to zero except for the constant term, which was set to 9. But in my approach, I used the identity sin^2 t + cos^2 t=1, which led me to the equation.Wait, perhaps both methods should lead to the same result, but in the initial problem, the user had:They started with ax^2 +bxy +cy^2=9, substituted x and y in terms of t, expanded, and then set coefficients of cos^2 t, sin t cos t, and sin^2 t to zero except for the constant term.Wait, but in my approach, I ended up with a=1, b=4/5, c=13/25, but in the initial problem, the user had a=1/9, b=4/45, c=4/225.Wait, that's a discrepancy. So, perhaps I made a mistake in my approach.Wait, let me go back to my steps.I had:x =3 cos t -2 sin ty=5 sin tExpressed sin t = y/5, cos t = x/3 + 2y/15Then, substituted into sin^2 t + cos^2 t=1, leading to:(y/5)^2 + (x/3 + 2y/15)^2=1Which expanded to:y^2/25 + x^2/9 +4xy/45 +4y^2/225=1Then, combining y^2 terms:9y^2/225 +4y^2/225=13y^2/225So, equation becomes:x^2/9 +4xy/45 +13y^2/225=1Multiply both sides by 225:25x^2 +20xy +13y^2=225Divide by 25:x^2 + (4/5)xy + (13/25)y^2=9So, a=1, b=4/5, c=13/25.But in the initial problem, the user had a different approach, leading to a=1/9, b=4/45, c=4/225.Wait, perhaps I made a mistake in the initial substitution.Wait, let me check the substitution again.From x=3 cos t -2 sin t, and y=5 sin t.Expressed as:x=3 cos t - (2/5)ySo, cos t=(x + (2/5)y)/3Wait, is that correct?Wait, x=3 cos t -2 sin tBut y=5 sin t, so sin t=y/5.Thus, x=3 cos t -2*(y/5)=3 cos t - (2y)/5So, 3 cos t =x + (2y)/5Thus, cos t=(x + (2y)/5)/3= x/3 + (2y)/15Which is what I had before.So, cos t= x/3 + 2y/15Thus, sin t= y/5So, sin^2 t + cos^2 t=1Which gives:(y/5)^2 + (x/3 + 2y/15)^2=1Which expands to:y^2/25 + x^2/9 +4xy/45 +4y^2/225=1Which is correct.Then, combining y^2 terms:9y^2/225 +4y^2/225=13y^2/225So, equation becomes:x^2/9 +4xy/45 +13y^2/225=1Multiply by 225:25x^2 +20xy +13y^2=225Divide by 25:x^2 + (4/5)xy + (13/25)y^2=9So, a=1, b=4/5, c=13/25.Wait, but in the initial problem, the user had a different result. Let me check their steps.In the initial problem, the user wrote:They had x=3 cos t -2 sin t, y=5 sin t.Then, they considered ax^2 +bxy +cy^2=9, substituted x and y in terms of t, expanded, and set coefficients to match 9.Wait, perhaps that's a different approach, but both methods should lead to the same result. Let me try that approach.So, starting with ax^2 +bxy +cy^2=9.Substitute x=3 cos t -2 sin t, y=5 sin t.Compute ax^2:a*(3 cos t -2 sin t)^2 =a*(9 cos^2 t -12 cos t sin t +4 sin^2 t)Compute bxy:b*(3 cos t -2 sin t)*(5 sin t)=b*(15 cos t sin t -10 sin^2 t)Compute cy^2:c*(5 sin t)^2=25c sin^2 tSo, adding all together:9a cos^2 t -12a cos t sin t +4a sin^2 t +15b cos t sin t -10b sin^2 t +25c sin^2 t=9Now, group like terms:cos^2 t: 9acos t sin t: (-12a +15b)sin^2 t: (4a -10b +25c)So, the equation becomes:9a cos^2 t + (-12a +15b) cos t sin t + (4a -10b +25c) sin^2 t=9Now, since this must hold for all t, the coefficients of cos^2 t, cos t sin t, and sin^2 t must be such that the equation equals 9 for all t. However, the left-hand side is a combination of cos^2 t, sin^2 t, and cos t sin t, which are functions that vary with t, while the right-hand side is a constant.The only way this can hold for all t is if the coefficients of cos^2 t, cos t sin t, and sin^2 t are zero, except for a constant term. But in our case, the right-hand side is 9, which is a constant. Therefore, the coefficients of cos^2 t, cos t sin t, and sin^2 t must be zero, and the constant term must be 9.Wait, but in our case, the left-hand side is entirely in terms of cos^2 t, sin^2 t, and cos t sin t, with no constant term. So, to make the entire expression equal to 9 for all t, we must have:9a =0-12a +15b=04a -10b +25c=0But wait, that would imply that 9a=0, which would mean a=0, but then the equation would be 0 +0 +0=9, which is impossible. So, that approach must be incorrect.Wait, perhaps the initial problem's approach is wrong because when substituting, the left-hand side becomes a combination of cos^2 t, sin^2 t, and cos t sin t, which cannot be a constant unless their coefficients are zero, but then the equation would be 0=9, which is impossible. Therefore, that approach is flawed.Wait, but in reality, the curve is a conic section, and the equation ax^2 +bxy +cy^2=9 should hold for all points (x,y) on the curve, which are parameterized by t. Therefore, the expression ax^2 +bxy +cy^2 must equal 9 for all t, which implies that the coefficients of cos^2 t, sin^2 t, and cos t sin t must be such that the entire expression is a constant 9, regardless of t.Therefore, the coefficients of cos^2 t, sin^2 t, and cos t sin t must satisfy certain conditions. Specifically, the coefficients must be such that the expression is a constant, which in this case is 9.So, let's denote:Let me denote:A =9aB= -12a +15bC=4a -10b +25cSo, the equation becomes:A cos^2 t + B cos t sin t + C sin^2 t=9But since this must hold for all t, we can use the identity that any expression of the form A cos^2 t + B cos t sin t + C sin^2 t can be written as a constant only if it's a multiple of the identity cos^2 t + sin^2 t=1, but with coefficients adjusted appropriately.Alternatively, we can express the left-hand side as a linear combination of cos^2 t, sin^2 t, and cos t sin t, and set it equal to 9.But since cos^2 t and sin^2 t are not constants, the only way for their combination to be a constant is if their coefficients are zero, and the remaining term is the constant.Wait, but in our case, the left-hand side is A cos^2 t + B cos t sin t + C sin^2 t, and we need this to equal 9 for all t.But cos^2 t and sin^2 t vary between 0 and 1, and cos t sin t varies between -0.5 and 0.5. Therefore, the only way for their combination to be a constant is if their coefficients are such that the entire expression is a constant.Wait, but how? Because cos^2 t + sin^2 t=1, but in our case, the coefficients are different.Wait, perhaps we can write the left-hand side as:A cos^2 t + B cos t sin t + C sin^2 t =9But since cos^2 t + sin^2 t=1, we can write:(A) cos^2 t + (C) sin^2 t + B cos t sin t =9But to make this equal to 9 for all t, we can consider that the expression must be equal to 9 regardless of t, which implies that the coefficients of cos^2 t and sin^2 t must be equal, and the coefficient of cos t sin t must be zero, because otherwise, the expression would vary with t.Wait, let me think. Suppose we have:A cos^2 t + C sin^2 t + B cos t sin t =9We can write this as:(A - C) cos^2 t + C (cos^2 t + sin^2 t) + B cos t sin t =9But cos^2 t + sin^2 t=1, so:(A - C) cos^2 t + C*1 + B cos t sin t =9So, (A - C) cos^2 t + B cos t sin t + C =9Now, for this to hold for all t, the coefficients of cos^2 t and cos t sin t must be zero, and the constant term must be 9.Therefore:A - C=0B=0C=9So, from A - C=0, we have A=CFrom B=0, we have B=0From C=9, we have C=9Therefore, A=9, B=0, C=9But A=9a, B=-12a +15b, C=4a -10b +25cSo, set up the equations:9a=9 => a=1-12a +15b=0 => -12*1 +15b=0 =>15b=12 =>b=12/15=4/54a -10b +25c=9 =>4*1 +10*(4/5) +25c=9 =>4 +8 +25c=9 =>12 +25c=9 =>25c= -3 =>c= -3/25Wait, that's different from my earlier result. Hmm, that can't be right because when I plugged in t=0, I had a=1, which worked, but now with c=-3/25, let's check.Wait, at t=0, x=3, y=0.Plug into the equation:a x^2 +b x y +c y^2=9Which is 1*(9) + (4/5)*(0) + (-3/25)*(0)=9, which is correct.At t=π/2, x=-2, y=5.Plug into the equation:1*(-2)^2 + (4/5)*(-2)*(5) + (-3/25)*(5)^2=4 + (4/5)*(-10) + (-3/25)*25=4 -8 -3= -7But this should equal 9, which it doesn't. So, that's a problem.Wait, so this approach leads to inconsistency. Therefore, perhaps the assumption that A - C=0 and B=0 is incorrect.Wait, maybe I made a mistake in the approach. Alternatively, perhaps the initial substitution method is the correct way, leading to a=1, b=4/5, c=13/25.Wait, let me check again.When I used the substitution method, I arrived at:x^2 + (4/5)xy + (13/25)y^2=9Which, when tested with t=0, t=π/2, and t=π/6, gave consistent results.On the other hand, when I tried the other approach, setting coefficients to make the expression equal to 9 for all t, I ended up with a=1, b=4/5, c=-3/25, which didn't work when tested.Therefore, perhaps the substitution method is correct, and the other approach was flawed.Wait, perhaps the issue is that in the second approach, I assumed that the expression must equal 9 for all t, but in reality, the expression ax^2 +bxy +cy^2=9 is supposed to represent the curve, which is parameterized by t. Therefore, for each t, (x(t), y(t)) lies on the curve, so ax^2 +bxy +cy^2 must equal 9 for all t.Therefore, the expression 9a cos^2 t + (-12a +15b) cos t sin t + (4a -10b +25c) sin^2 t must equal 9 for all t.But since this is a combination of cos^2 t, sin^2 t, and cos t sin t, which are functions that vary with t, the only way for their combination to be a constant 9 for all t is if the coefficients of cos^2 t and sin^2 t are equal, and the coefficient of cos t sin t is zero, and the constant term is 9.Wait, but in our case, the expression is:9a cos^2 t + (-12a +15b) cos t sin t + (4a -10b +25c) sin^2 t=9To make this equal to 9 for all t, we can use the identity that any quadratic form in cos t and sin t can be expressed as a constant only if it's a multiple of cos^2 t + sin^2 t=1.Therefore, we can set:9a = k4a -10b +25c =k-12a +15b=0And then, since 9a cos^2 t + (4a -10b +25c) sin^2 t + (-12a +15b) cos t sin t=9But since cos^2 t + sin^2 t=1, we can write:k (cos^2 t + sin^2 t) + (-12a +15b) cos t sin t=9Which simplifies to:k*1 + (-12a +15b) cos t sin t=9Therefore, for this to hold for all t, the coefficient of cos t sin t must be zero, and k must equal 9.Therefore:-12a +15b=0k=9But k=9a=9, so a=1From -12a +15b=0, with a=1:-12 +15b=0 =>15b=12 =>b=12/15=4/5Now, from k=9a=9, and k=4a -10b +25c=9So, 4a -10b +25c=9Substitute a=1, b=4/5:4*1 -10*(4/5) +25c=94 -8 +25c=9-4 +25c=925c=13c=13/25Therefore, a=1, b=4/5, c=13/25Which matches the result from the substitution method.So, the correct ordered triple is (1, 4/5, 13/25).Wait, but in the initial problem, the user had a different answer. Let me check their steps again.In the initial problem, the user wrote:They had:ax^2 +bxy +cy^2=9Substituted x=3 cos t -2 sin t, y=5 sin tExpanded to:9a cos^2 t + (-12a +15b) cos t sin t + (4a -10b +25c) sin^2 t=9Then, they set:9a=1-12a +15b=04a -10b +25c=0But this leads to:a=1/9b= (12a)/15= (12/9)/15= (4/3)/15=4/45c= (10b -4a)/25= (10*(4/45) -4*(1/9))/25= (40/45 -4/9)/25= (40/45 -20/45)/25=20/45/25=4/225But this leads to a=1/9, b=4/45, c=4/225But when I tested this with t=0, x=3, y=0:a x^2 +b x y +c y^2= (1/9)*9 + (4/45)*0 + (4/225)*0=1 +0+0=1≠9Which is incorrect.Therefore, the initial problem's approach was incorrect because they set 9a=1, which is wrong. Instead, we should set 9a=9, leading to a=1, and then proceed accordingly.So, the correct answer is a=1, b=4/5, c=13/25.Therefore, the ordered triple is (1, 4/5, 13/25).
❓Consider a crank-slider mechanism (1 - crank, 2 - connecting rod). They were first used in antiquity in Roman sawmills, where the rotation of a wheel driven by falling water was converted into the reciprocating motion of the saw blade. This crank rotates with a constant angular velocity omega = 10 rad/s. Find the equations of motion, the equation of the trajectory, and the velocity of point M on the connecting rod, given O A = A B = 90 , text{cm}, and A M = frac{1}{2} A B.
💡Okay, so I need to figure out the equations of motion, the trajectory, and the velocity of point M on the connecting rod in this crank-slider mechanism. Let me start by understanding the problem.First, the crank-slider mechanism has a crank (OA) and a connecting rod (AB). Both OA and AB are 90 cm long. Point M is located halfway between A and B, so AM is half of AB, which means AM is 45 cm.The crank is rotating with a constant angular velocity ω = 10 rad/s. I need to find the position of point M as a function of time, its trajectory, and its velocity.Let me visualize this. The crank OA is rotating around point O, which is fixed. As OA rotates, point A moves in a circular path. The connecting rod AB is attached to point A and slides along a straight path, converting the rotational motion into linear motion.Since OA is 90 cm, point A's position can be described using trigonometric functions. If I consider the angle θ that OA makes with the horizontal axis at any time t, then θ = ωt because the angular velocity is constant.So, the coordinates of point A at time t would be:- x-coordinate: OA * cos(θ) = 90 * cos(10t)- y-coordinate: OA * sin(θ) = 90 * sin(10t)Now, point B is the other end of the connecting rod AB, which is also 90 cm. Since AB is connected to point A and slides along a straight path, the position of point B depends on the position of point A.But wait, in a crank-slider mechanism, point B is usually constrained to move along a straight line, often the x-axis. So, point B can only move horizontally. That means the y-coordinate of point B is zero.Given that AB is 90 cm, the position of point B can be found using the coordinates of point A and the fact that AB is 90 cm. Let me denote the coordinates of point B as (x_B, 0).Using the distance formula between points A and B:√[(x_B - x_A)^2 + (0 - y_A)^2] = AB = 90 cmPlugging in the coordinates of A:√[(x_B - 90 cos(10t))^2 + (0 - 90 sin(10t))^2] = 90Squaring both sides:(x_B - 90 cos(10t))^2 + (90 sin(10t))^2 = 8100Expanding the first term:(x_B^2 - 180 x_B cos(10t) + 8100 cos²(10t)) + 8100 sin²(10t) = 8100Combine like terms:x_B^2 - 180 x_B cos(10t) + 8100 (cos²(10t) + sin²(10t)) = 8100Since cos² + sin² = 1:x_B^2 - 180 x_B cos(10t) + 8100 = 8100Subtract 8100 from both sides:x_B^2 - 180 x_B cos(10t) = 0Factor out x_B:x_B (x_B - 180 cos(10t)) = 0So, x_B = 0 or x_B = 180 cos(10t)But x_B = 0 would mean point B is at the origin, which isn't the case because AB is 90 cm. So, x_B = 180 cos(10t)Wait, that doesn't make sense because AB is 90 cm, not 180 cm. Did I make a mistake?Let me check the distance formula again. The distance between A and B should be 90 cm, so:√[(x_B - x_A)^2 + (0 - y_A)^2] = 90Plugging in x_A = 90 cos(10t) and y_A = 90 sin(10t):√[(x_B - 90 cos(10t))^2 + (90 sin(10t))^2] = 90Square both sides:(x_B - 90 cos(10t))^2 + (90 sin(10t))^2 = 8100Expand:x_B^2 - 180 x_B cos(10t) + 8100 cos²(10t) + 8100 sin²(10t) = 8100Combine cos² and sin²:x_B^2 - 180 x_B cos(10t) + 8100 (cos²(10t) + sin²(10t)) = 8100Which simplifies to:x_B^2 - 180 x_B cos(10t) + 8100 = 8100Subtract 8100:x_B^2 - 180 x_B cos(10t) = 0Factor:x_B (x_B - 180 cos(10t)) = 0So, x_B = 0 or x_B = 180 cos(10t)But x_B can't be 180 cos(10t) because AB is only 90 cm. Hmm, maybe I need to reconsider.Wait, perhaps I made a mistake in assuming the position of point B. In a crank-slider mechanism, the slider (point B) moves along a straight path, usually the x-axis, but the connecting rod AB is not necessarily aligned with the crank OA. So, the position of point B is determined by the position of point A and the length of AB.Let me try a different approach. Let's denote the coordinates of point B as (x_B, 0). The vector from A to B is (x_B - x_A, -y_A). The length of this vector is AB = 90 cm.So, the vector AB has magnitude 90 cm, so:√[(x_B - x_A)^2 + (0 - y_A)^2] = 90Which is the same as before. So, plugging in x_A = 90 cos(10t) and y_A = 90 sin(10t):√[(x_B - 90 cos(10t))^2 + (90 sin(10t))^2] = 90Square both sides:(x_B - 90 cos(10t))^2 + (90 sin(10t))^2 = 8100Expand:x_B^2 - 180 x_B cos(10t) + 8100 cos²(10t) + 8100 sin²(10t) = 8100Again, cos² + sin² = 1:x_B^2 - 180 x_B cos(10t) + 8100 = 8100So, x_B^2 - 180 x_B cos(10t) = 0Factor:x_B (x_B - 180 cos(10t)) = 0So, x_B = 0 or x_B = 180 cos(10t)But x_B = 180 cos(10t) would mean the slider moves between -180 cm and 180 cm, but the connecting rod is only 90 cm. That doesn't make sense. Maybe I need to consider that the slider's position is constrained such that the connecting rod doesn't extend beyond the crank's rotation.Wait, perhaps the slider's position is not x_B = 180 cos(10t), but rather x_B = OA cos(θ) + AB cos(φ), where φ is the angle of the connecting rod. But I'm not sure.Alternatively, maybe I should use the law of cosines to find the position of point B.In triangle OAB, OA = 90 cm, AB = 90 cm, and OB is the distance from O to B, which is x_B since B is on the x-axis.Using the law of cosines:OB² = OA² + AB² - 2 * OA * AB * cos(angle OAB)But I don't know angle OAB. Hmm.Alternatively, since OA and AB are both 90 cm, triangle OAB is isoceles with OA = AB. So, angle at A is equal to angle at O.Wait, but point B is constrained to move along the x-axis, so triangle OAB is not necessarily isoceles in terms of angles.This is getting confusing. Maybe I should use parametric equations.Let me denote the angle of OA as θ = 10t.Then, coordinates of A are (90 cos θ, 90 sin θ).Point B is (x_B, 0).The distance AB is 90 cm, so:√[(x_B - 90 cos θ)^2 + (0 - 90 sin θ)^2] = 90Square both sides:(x_B - 90 cos θ)^2 + (90 sin θ)^2 = 8100Expand:x_B² - 180 x_B cos θ + 8100 cos² θ + 8100 sin² θ = 8100Again, cos² θ + sin² θ = 1:x_B² - 180 x_B cos θ + 8100 = 8100So, x_B² - 180 x_B cos θ = 0Factor:x_B (x_B - 180 cos θ) = 0Thus, x_B = 0 or x_B = 180 cos θBut x_B = 180 cos θ would mean the slider moves between -180 cm and 180 cm, which is twice the length of AB. That doesn't seem right.Wait, maybe I'm misunderstanding the mechanism. In a typical crank-slider, the connecting rod is longer than the crank. If OA and AB are both 90 cm, then it's a special case where the connecting rod is equal in length to the crank.In that case, the maximum displacement of the slider would be OA + AB = 180 cm, which matches the x_B = 180 cos θ result. So, maybe it is correct.So, point B moves between -180 cm and 180 cm along the x-axis.Now, point M is halfway between A and B, so AM = 45 cm.So, the coordinates of M can be found by averaging the coordinates of A and B.So, M_x = (x_A + x_B)/2 = (90 cos θ + 180 cos θ)/2 = (270 cos θ)/2 = 135 cos θSimilarly, M_y = (y_A + y_B)/2 = (90 sin θ + 0)/2 = 45 sin θWait, that can't be right because if M is halfway between A and B, and B is on the x-axis, then M's y-coordinate should be half of A's y-coordinate.But let's check:If M is the midpoint of AB, then:M_x = (x_A + x_B)/2M_y = (y_A + y_B)/2Since y_B = 0, M_y = y_A / 2 = 45 sin θAnd x_B = 180 cos θ, so M_x = (90 cos θ + 180 cos θ)/2 = (270 cos θ)/2 = 135 cos θSo, M has coordinates (135 cos θ, 45 sin θ)But wait, that would mean M is moving in a circular path with radius 135 cm in x and 45 cm in y. That doesn't seem right because the connecting rod is only 90 cm.Wait, maybe I made a mistake in assuming M is the midpoint of AB. The problem says AM = 1/2 AB, so AM = 45 cm, which is correct. So, M divides AB into two parts: AM = 45 cm and MB = 45 cm.So, M is indeed the midpoint of AB.But then, as calculated, M_x = 135 cos θ and M_y = 45 sin θ.Wait, but if M is moving in a circular path with x = 135 cos θ and y = 45 sin θ, that's an ellipse, not a circle.Yes, because the x and y amplitudes are different. So, the trajectory of M is an ellipse.But let me confirm:If M_x = 135 cos θ and M_y = 45 sin θ, then the trajectory is indeed an ellipse with semi-major axis 135 cm and semi-minor axis 45 cm.But wait, in reality, point M is on the connecting rod, which is constrained by the slider. So, maybe the trajectory is not an ellipse but something else.Wait, no, because as the crank rotates, point M moves in a circular motion relative to point A, but since point A is moving in a circle and point B is moving along a straight line, the combination results in an elliptical motion for M.So, the equations of motion for M are:x = 135 cos(10t)y = 45 sin(10t)Now, for the velocity, we need to find the derivatives of x and y with respect to time.So, velocity components:v_x = dx/dt = -135 * 10 sin(10t) = -1350 sin(10t) cm/sv_y = dy/dt = 45 * 10 cos(10t) = 450 cos(10t) cm/sSo, the velocity vector is (-1350 sin(10t), 450 cos(10t)) cm/s.Wait, but earlier I thought M was the midpoint, so maybe I should double-check the coordinates.If M is the midpoint, then:M_x = (x_A + x_B)/2M_y = (y_A + y_B)/2We have x_A = 90 cos θ, y_A = 90 sin θx_B = 180 cos θ, y_B = 0So, M_x = (90 cos θ + 180 cos θ)/2 = 135 cos θM_y = (90 sin θ + 0)/2 = 45 sin θYes, that's correct.So, the equations of motion are:x = 135 cos(10t)y = 45 sin(10t)And the velocity components are:v_x = -1350 sin(10t)v_y = 450 cos(10t)But wait, the problem states OA = AB = 90 cm, and AM = 1/2 AB, so AM = 45 cm. So, M is indeed the midpoint.But I'm a bit confused because in a typical crank-slider, the connecting rod is longer than the crank, but here they are equal. So, the slider's maximum displacement is 180 cm, which is twice the crank length.But regardless, the calculations seem consistent.So, to summarize:Equations of motion:x(t) = 135 cos(10t)y(t) = 45 sin(10t)Trajectory: This is an ellipse given by (x/135)^2 + (y/45)^2 = cos²(10t) + sin²(10t) = 1So, the trajectory equation is (x/135)^2 + (y/45)^2 = 1Velocity components:v_x = -1350 sin(10t)v_y = 450 cos(10t)So, the velocity vector is (-1350 sin(10t), 450 cos(10t)) cm/s.Wait, but let me check the units. Angular velocity ω is given in rad/s, so the derivatives will have units of cm/s.Yes, that's correct.But I'm still a bit unsure about the trajectory being an ellipse. Let me think again.If M is moving such that x = 135 cos θ and y = 45 sin θ, then indeed, (x/135)^2 + (y/45)^2 = cos² θ + sin² θ = 1, which is the equation of an ellipse.So, yes, the trajectory is an ellipse.Therefore, the final answers are:Equations of motion:x(t) = 135 cos(10t)y(t) = 45 sin(10t)Trajectory:(x/135)^2 + (y/45)^2 = 1Velocity components:v_x = -1350 sin(10t)v_y = 450 cos(10t)So, the velocity vector is (-1350 sin(10t), 450 cos(10t)) cm/s.I think that's it.
❓Given a, b, c in mathbb{R}, the statement "b^2 - 4ac < 0" is a condition for "the inequality ax^2 + bx + c < 0 always holds over mathbb{R}" is ( )A: Sufficient but not necessary conditionB: Necessary but not sufficient conditionC: Necessary and sufficient conditionD: Neither sufficient nor necessary condition
💡Okay, so I have this problem here about quadratic inequalities. It says, given real numbers a, b, and c, the statement "b² - 4ac < 0" is a condition for "the inequality ax² + bx + c < 0 always holds over ℝ." Then it asks whether this condition is sufficient, necessary, both, or neither.Alright, let me recall what I know about quadratic inequalities. A quadratic equation ax² + bx + c = 0 has solutions depending on the discriminant, which is b² - 4ac. If the discriminant is less than zero, that means there are no real roots, right? So the quadratic doesn't cross the x-axis.Now, if the quadratic doesn't cross the x-axis, its graph is either entirely above or entirely below the x-axis. Whether it's above or below depends on the coefficient 'a'. If a is positive, the parabola opens upwards, so if there are no real roots, the entire graph is above the x-axis, meaning ax² + bx + c > 0 for all x. Conversely, if a is negative, the parabola opens downwards, so the entire graph is below the x-axis, meaning ax² + bx + c < 0 for all x.So, the discriminant being less than zero (b² - 4ac < 0) tells us that the quadratic doesn't cross the x-axis, but it doesn't tell us whether it's always positive or always negative. That depends on the sign of 'a'. Therefore, if someone says that b² - 4ac < 0 is a condition for ax² + bx + c < 0 always holding, that's only part of the story. We also need to know that a is negative. Without knowing the sign of 'a', we can't be sure whether the quadratic is always positive or always negative.So, is b² - 4ac < 0 a sufficient condition? Well, if we have b² - 4ac < 0, does that guarantee that ax² + bx + c < 0 for all x? No, because if a is positive, it would be greater than zero. So it's not sufficient on its own.Is it a necessary condition? That is, if ax² + bx + c < 0 for all x, does that necessarily mean that b² - 4ac < 0? Yes, because if the quadratic is always negative, it can't cross the x-axis, so the discriminant must be negative. So it's a necessary condition.Wait, but the question is phrased as "the statement 'b² - 4ac < 0' is a condition for 'the inequality ax² + bx + c < 0 always holds over ℝ'". So, is the discriminant condition a sufficient or necessary condition for the inequality to hold?From what I just thought, it's necessary but not sufficient. Because without knowing the sign of 'a', we can't guarantee the inequality holds just from the discriminant. But if the inequality does hold, then the discriminant must be negative.But hold on, the options are:A: Sufficient but not necessary conditionB: Necessary but not sufficient conditionC: Necessary and sufficient conditionD: Neither sufficient nor necessary conditionSo, based on my reasoning, it's necessary but not sufficient. So the answer should be B.Wait, but let me double-check. If I have b² - 4ac < 0, does that mean the inequality always holds? No, because if a is positive, it doesn't. So it's not sufficient. But if the inequality does always hold, then b² - 4ac must be less than zero. So it's necessary.Therefore, the condition is necessary but not sufficient. So B is the correct answer.But wait, in the initial problem statement, it's phrased as "the statement 'b² - 4ac < 0' is a condition for 'the inequality ax² + bx + c < 0 always holds over ℝ'". So, is the discriminant condition a condition for the inequality? If someone says "if b² - 4ac < 0, then the inequality holds", that's not true because we need a < 0 as well. So the discriminant condition alone is not sufficient.But if someone says "for the inequality to hold, it is necessary that b² - 4ac < 0", that's true. So the discriminant condition is necessary but not sufficient.So yeah, I think B is the correct answer.Wait, but in the initial problem, the user wrote the same problem twice. Maybe that's a typo. But regardless, the reasoning remains the same.So, to summarize:- Discriminant < 0 is necessary for the inequality to hold always, because if the inequality holds always, the quadratic doesn't cross the x-axis, so discriminant must be negative.- However, discriminant < 0 is not sufficient because we also need a < 0 to ensure the quadratic is always negative.Therefore, the condition is necessary but not sufficient, so the answer is B.**Final Answer**boxed{B}