💡First, I need to determine the amount of loss incurred by subtracting the selling price from the cost price.Next, I'll calculate the percentage of loss by dividing the loss by the cost price and then multiplying by 100 to get the percentage.Finally, I'll present the calculated percentage of loss as the solution.

💡Okay, so I have this problem here about an ellipse and finding the range of values for the expression x + 2y. Let me try to figure this out step by step. First, the equation of the ellipse is given as (x²)/4 + (y²)/3 = 1. I remember that the standard form of an ellipse is (x²)/a² + (y²)/b² = 1, where a and b are the semi-major and semi-minor axes. So in this case, a² is 4 and b² is 3, which means a is 2 and b is sqrt(3). Now, the question is asking for the range of x + 2y. Hmm, that sounds like a linear expression, and I think I need to find the maximum and minimum values that this expression can take given that (x, y) lies on the ellipse. I recall that for such optimization problems on ellipses, one common method is to use parametric equations. So, I can parameterize the ellipse using trigonometric functions. Since a is 2 and b is sqrt(3), I can write x as 2 cos θ and y as sqrt(3) sin θ, where θ is the parameter varying from 0 to 2π. Let me write that down:x = 2 cos θy = sqrt(3) sin θNow, substituting these into the expression x + 2y, we get:x + 2y = 2 cos θ + 2 * sqrt(3) sin θSimplify that:x + 2y = 2 cos θ + 2 sqrt(3) sin θHmm, this looks like a combination of sine and cosine terms. I think I can express this as a single sine or cosine function using the amplitude-phase form. The general formula is A cos θ + B sin θ = C sin(θ + φ), where C is the amplitude and φ is the phase shift. Let me calculate the amplitude C. The formula for C is sqrt(A² + B²). Here, A is 2 and B is 2 sqrt(3). So, C = sqrt(2² + (2 sqrt(3))²) = sqrt(4 + 12) = sqrt(16) = 4. So, the expression 2 cos θ + 2 sqrt(3) sin θ can be rewritten as 4 sin(θ + φ) for some phase φ. But wait, actually, it can be either sine or cosine, depending on how we set it up. Let me double-check. The formula is A cos θ + B sin θ = C cos(θ - φ), where C = sqrt(A² + B²) and tan φ = B/A. Alternatively, it can also be written as C sin(θ + φ). Either way, the amplitude is the same, which is 4. So, regardless of the phase shift, the maximum value of this expression will be 4 and the minimum will be -4 because the sine and cosine functions oscillate between -1 and 1. Therefore, the range of x + 2y is from -4 to 4. Let me just verify this another way to be sure. Maybe using Lagrange multipliers or something. Alternatively, I can think of x + 2y as a linear function and find its maximum and minimum on the ellipse. This is essentially finding the extrema of the linear function subject to the ellipse constraint. Using the method of Lagrange multipliers, we set up the gradients. The gradient of x + 2y is (1, 2), and the gradient of the ellipse equation is (x/2, 2y/3). Setting them proportional:1 = λ*(x/2)2 = λ*(2y/3)From the first equation: λ = 2/xFrom the second equation: 2 = (2λ y)/3 => 2 = (2*(2/x)*y)/3 => 2 = (4y)/(3x) => 6x = 4y => 3x = 2y => y = (3/2)xNow, substitute y = (3/2)x into the ellipse equation:(x²)/4 + (( (3/2)x )²)/3 = 1Simplify:(x²)/4 + (9x²/4)/3 = 1(x²)/4 + (3x²)/4 = 1(4x²)/4 = 1x² = 1x = ±1So, when x = 1, y = (3/2)(1) = 3/2When x = -1, y = (3/2)(-1) = -3/2Now, compute x + 2y for these points:For (1, 3/2): 1 + 2*(3/2) = 1 + 3 = 4For (-1, -3/2): -1 + 2*(-3/2) = -1 - 3 = -4So, the maximum value is 4 and the minimum is -4. That confirms my earlier result. Therefore, the range of x + 2y is [-4, 4], which corresponds to option C.**Final Answer**boxed{C}

💡Okay, so I have this problem here: "What number must we subtract from 5 - 3i to get -1 + 4i?" Hmm, let's see. I'm just starting to learn about complex numbers, so I need to take it step by step.First, I know that complex numbers have a real part and an imaginary part. In this case, 5 - 3i has a real part of 5 and an imaginary part of -3i. The result we want is -1 + 4i, which has a real part of -1 and an imaginary part of 4i.The question is asking what number we need to subtract from 5 - 3i to get -1 + 4i. Let me denote that unknown number as z. So, mathematically, this can be written as:5 - 3i - z = -1 + 4iI need to solve for z. To do that, I can rearrange the equation to isolate z. Let me move z to the other side and the constants to the left:5 - 3i + 1 - 4i = zWait, is that right? Let me double-check. If I have 5 - 3i - z = -1 + 4i, then to solve for z, I can add z to both sides and add 1 - 4i to both sides. So:5 - 3i + 1 - 4i = zNow, combining like terms. The real parts are 5 and 1, so 5 + 1 is 6. The imaginary parts are -3i and -4i, so -3i - 4i is -7i. Therefore, z should be 6 - 7i.But wait, let me make sure I didn't make a mistake. If I subtract 6 - 7i from 5 - 3i, do I get -1 + 4i?Let's check:(5 - 3i) - (6 - 7i) = 5 - 3i - 6 + 7iCombining real parts: 5 - 6 = -1Combining imaginary parts: -3i + 7i = 4iSo, yes, it does equal -1 + 4i. That seems correct.I think I got it right. The number to subtract is 6 - 7i.

💡First, I need to understand what the problem is asking. It wants me to solve for ( n ) in the equation ( 0.04n + 0.1(30 + n) = 15.2 ), but with a 10% increase applied to ( n ). This means that the value of ( n ) I find should be 10% higher than the original value.To incorporate the 10% increase, I'll introduce a new variable ( m ) that represents ( n ) after the increase. So, ( m = 1.1n ). This substitution allows me to rewrite the original equation in terms of ( m ), making it easier to solve for ( n ) after considering the increase.Substituting ( n ) with ( frac{m}{1.1} ) in the equation, I get:[0.04 left( frac{m}{1.1} right) + 0.1 left( 30 + frac{m}{1.1} right) = 15.2]Next, I'll simplify the equation by distributing and combining like terms. This involves multiplying through by 1.1 to eliminate the denominator and then isolating ( m ) to find its value.After solving for ( m ), I'll use the relationship ( m = 1.1n ) to find the original value of ( n ) before the 10% increase. This will give me the final value of ( n ) that satisfies the equation with the applied increase.

💡First, I need to determine the amount of plankton the whale consumes each hour. The problem states that the whale's consumption increases by a constant amount each hour, which suggests an arithmetic sequence.Let ( x ) represent the amount of plankton consumed in the first hour, and let ( y ) represent the constant increase in consumption each subsequent hour.The total consumption over 9 hours can be calculated using the formula for the sum of an arithmetic series:[text{Total} = frac{n}{2} times (2a + (n - 1)d)]where ( n = 9 ), ( a = x ), and ( d = y ).Given that the total consumption is 270 kilos, the equation becomes:[frac{9}{2} times (2x + 8y) = 270]Simplifying this:[9x + 36y = 270]Additionally, the consumption on the sixth hour is given as 33 kilos. The sixth hour's consumption can be expressed as:[x + 5y = 33]Now, I have two equations:1. ( 9x + 36y = 270 )2. ( x + 5y = 33 )To solve these equations, I'll use the substitution method. From the second equation:[x = 33 - 5y]Substituting ( x ) into the first equation:[9(33 - 5y) + 36y = 270][297 - 45y + 36y = 270][-9y = -27][y = 3]Thus, the whale consumes 3 kilos more of plankton each hour than the previous hour.

💡Okay, so I have this complex number z, and I need to find the imaginary part of its complex conjugate, which is denoted as (bar{z}). The expression for z is given as (i^{2018} - frac{|3 - 4i|}{3 - 4i}). Hmm, that looks a bit complicated, but I think I can break it down step by step.First, let me recall some properties of complex numbers. I know that (i) is the imaginary unit, where (i^2 = -1). Also, the modulus of a complex number (a + bi) is given by (|a + bi| = sqrt{a^2 + b^2}). The complex conjugate of (a + bi) is (a - bi). So, the complex conjugate changes the sign of the imaginary part.Now, looking at the expression for z, it has two main parts: (i^{2018}) and (frac{|3 - 4i|}{3 - 4i}). I need to simplify each part separately and then combine them.Starting with (i^{2018}). Since (i) has a cyclicity of 4, meaning that the powers of (i) repeat every four exponents. So, (i^1 = i), (i^2 = -1), (i^3 = -i), (i^4 = 1), and then (i^5 = i) again, and so on. Therefore, to simplify (i^{2018}), I can divide 2018 by 4 and find the remainder to determine which power of (i) it corresponds to.Calculating (2018 div 4), I get 504 with a remainder of 2 because (4 times 504 = 2016), and (2018 - 2016 = 2). So, (i^{2018} = i^2 = -1). That simplifies the first part of z to -1.Next, I need to simplify the second part: (frac{|3 - 4i|}{3 - 4i}). Let's first find the modulus of (3 - 4i). The modulus is calculated as (|3 - 4i| = sqrt{3^2 + (-4)^2} = sqrt{9 + 16} = sqrt{25} = 5). So, the numerator is 5.Now, the expression becomes (frac{5}{3 - 4i}). To simplify this, I need to eliminate the imaginary unit from the denominator. I can do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of (3 - 4i) is (3 + 4i).So, multiplying numerator and denominator by (3 + 4i):[frac{5}{3 - 4i} times frac{3 + 4i}{3 + 4i} = frac{5(3 + 4i)}{(3 - 4i)(3 + 4i)}]Calculating the denominator first: ((3 - 4i)(3 + 4i)). This is a difference of squares, so it simplifies to (3^2 - (4i)^2 = 9 - 16i^2). Since (i^2 = -1), this becomes (9 - 16(-1) = 9 + 16 = 25).So, the denominator is 25. Now, the numerator is (5(3 + 4i) = 15 + 20i). Therefore, the entire expression simplifies to:[frac{15 + 20i}{25} = frac{15}{25} + frac{20i}{25} = frac{3}{5} + frac{4}{5}i]So, the second part of z is (frac{3}{5} + frac{4}{5}i).Now, putting it all together, z is:[z = i^{2018} - frac{|3 - 4i|}{3 - 4i} = -1 - left( frac{3}{5} + frac{4}{5}i right)]Subtracting the complex number (frac{3}{5} + frac{4}{5}i) from -1, I need to distribute the negative sign:[z = -1 - frac{3}{5} - frac{4}{5}i]Combining the real parts:[-1 - frac{3}{5} = -frac{5}{5} - frac{3}{5} = -frac{8}{5}]So, z simplifies to:[z = -frac{8}{5} - frac{4}{5}i]Now, I need to find the complex conjugate of z, denoted as (bar{z}). The complex conjugate of a complex number (a + bi) is (a - bi). Applying this to z:[bar{z} = -frac{8}{5} + frac{4}{5}i]The question asks for the imaginary part of (bar{z}). The imaginary part is the coefficient of (i), which in this case is (frac{4}{5}).So, the imaginary part of the complex conjugate of z is (frac{4}{5}).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning:1. Simplified (i^{2018}) correctly to -1.2. Calculated the modulus of (3 - 4i) as 5, which is correct.3. Multiplied numerator and denominator by the conjugate (3 + 4i) to get (frac{15 + 20i}{25}), which simplifies to (frac{3}{5} + frac{4}{5}i). That seems right.4. Subtracted this from -1: (-1 - frac{3}{5} - frac{4}{5}i = -frac{8}{5} - frac{4}{5}i). That looks correct.5. Took the complex conjugate by changing the sign of the imaginary part: (-frac{8}{5} + frac{4}{5}i). So, the imaginary part is (frac{4}{5}).Yes, everything checks out. I think I did it correctly.Another thing I could consider is verifying the modulus calculation. (|3 - 4i| = sqrt{3^2 + (-4)^2} = sqrt{9 + 16} = sqrt{25} = 5). That's correct.Also, when multiplying by the conjugate, I correctly expanded the denominator as (9 - (4i)^2 = 9 - (-16) = 25), which is accurate.So, I feel confident that my answer is correct.**Final Answer**boxed{D}