💡Okay, so I have this problem where there are 20 numbers arranged in a circle. It says that the sum of any six consecutive numbers is 24. I need to find out what number is in the 12th position if the number in the 1st position is 1. Hmm, let me try to figure this out step by step.First, since the numbers are arranged in a circle, the sequence wraps around. That means the 20th number is next to the 1st number. So, the arrangement is cyclical. Now, the key information is that the sum of any six consecutive numbers is 24. That seems like a repeating pattern because if every set of six numbers adds up to the same total, there must be some kind of periodicity or repetition in the numbers.Let me denote the numbers as ( a_1, a_2, a_3, ldots, a_{20} ). Given that ( a_1 = 1 ), I need to find ( a_{12} ).Since the sum of any six consecutive numbers is 24, let's write down a few of these sums:1. ( a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 24 )2. ( a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = 24 )3. ( a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 24 )4. And so on, up to the 20th number.If I subtract the first equation from the second, I get:( (a_2 + a_3 + a_4 + a_5 + a_6 + a_7) - (a_1 + a_2 + a_3 + a_4 + a_5 + a_6) = 24 - 24 )Simplifying this, most terms cancel out:( a_7 - a_1 = 0 )So, ( a_7 = a_1 ). Since ( a_1 = 1 ), that means ( a_7 = 1 ).Similarly, if I subtract the second equation from the third:( (a_3 + a_4 + a_5 + a_6 + a_7 + a_8) - (a_2 + a_3 + a_4 + a_5 + a_6 + a_7) = 24 - 24 )Again, simplifying:( a_8 - a_2 = 0 )So, ( a_8 = a_2 ).Continuing this pattern, I can see that each number six positions apart is equal. So, ( a_{i} = a_{i+6} ) for all ( i ).Given that there are 20 numbers, and 20 divided by 6 is not an integer, this suggests that the sequence repeats every 6 numbers, but since 6 and 20 are not multiples of each other, the pattern must be such that the numbers repeat every 2 numbers. Wait, let me think about that.If ( a_i = a_{i+6} ), then after 6 steps, the numbers repeat. But since the circle has 20 numbers, which is not a multiple of 6, the repetition must occur in a way that the entire circle is covered. So, perhaps the numbers repeat every 2 numbers? Let me check.If ( a_1 = a_7 = a_{13} = a_{19} ), and ( a_2 = a_8 = a_{14} = a_{20} ), and so on, then each number is equal to the number 6 positions ahead. Since 6 and 20 have a greatest common divisor of 2, this would mean that the sequence repeats every 2 numbers. So, the numbers alternate between two values.Let me denote the two values as ( x ) and ( y ). So, the sequence would be ( x, y, x, y, x, y, ldots ).Given that ( a_1 = x = 1 ), then ( a_2 = y ), ( a_3 = x = 1 ), ( a_4 = y ), and so on.Now, the sum of any six consecutive numbers is 24. Let's calculate the sum of six numbers in this alternating sequence:( x + y + x + y + x + y = 3x + 3y = 24 )Simplifying, we get:( 3(x + y) = 24 )So,( x + y = 8 )Since ( x = 1 ), then:( 1 + y = 8 )Therefore,( y = 7 )So, the sequence alternates between 1 and 7. That means all odd positions are 1, and all even positions are 7.Now, the 12th position is an even position, so ( a_{12} = 7 ).Wait, let me double-check this. If the sequence is 1,7,1,7,... then positions 1,3,5,... are 1, and positions 2,4,6,... are 7. So, position 12 is even, so it should be 7. That makes sense.But just to be thorough, let me verify the sum of six consecutive numbers. Let's take positions 1 to 6:1,7,1,7,1,7. The sum is 1+7+1+7+1+7 = 24. That works.Another set, positions 2 to 7:7,1,7,1,7,1. The sum is 7+1+7+1+7+1 = 24. That also works.And positions 12 to 17:Since position 12 is 7, then 12:7,13:1,14:7,15:1,16:7,17:1. Sum is 7+1+7+1+7+1 = 24. Perfect.So, everything checks out. The 12th position is indeed 7.

💡Okay, so I have this equation to solve: log base e of (a squared minus 8a plus 20) equals 3. Hmm, I remember that log base e is the natural logarithm, which is written as ln. So, the equation is ln(a² - 8a + 20) = 3. First, I need to recall how to solve logarithmic equations. I think the general approach is to rewrite the logarithmic equation in its exponential form. Since it's a natural logarithm, that means the base is e. So, if ln(x) = y, then x = e^y. Applying that here, I can rewrite the equation as:a² - 8a + 20 = e³I remember that e is approximately 2.718, so e³ is roughly 20.085. But for simplicity, maybe I can just keep it as e³ for now. So, the equation becomes:a² - 8a + 20 = e³Wait, but e³ is a specific number, so I can subtract e³ from both sides to set the equation to zero:a² - 8a + 20 - e³ = 0But maybe it's better to just move everything to one side:a² - 8a + (20 - e³) = 0Now, this looks like a quadratic equation in terms of a. The standard form of a quadratic equation is ax² + bx + c = 0, so in this case, the coefficients are:A = 1 (coefficient of a²)B = -8 (coefficient of a)C = 20 - e³ (constant term)To solve for a, I can use the quadratic formula, which is:a = [-B ± √(B² - 4AC)] / (2A)Plugging in the values:a = [8 ± √((-8)² - 4*1*(20 - e³))] / 2Simplify inside the square root:First, calculate (-8)², which is 64.Then, calculate 4*1*(20 - e³) which is 4*(20 - e³) = 80 - 4e³So, the discriminant becomes:64 - (80 - 4e³) = 64 - 80 + 4e³ = -16 + 4e³So, the equation now is:a = [8 ± √(-16 + 4e³)] / 2I can factor out a 4 from inside the square root:√(-16 + 4e³) = √[4(e³ - 4)] = 2√(e³ - 4)So, substituting back:a = [8 ± 2√(e³ - 4)] / 2I can factor out a 2 in the numerator:a = [2*(4 ± √(e³ - 4))] / 2 = 4 ± √(e³ - 4)So, the solutions are:a = 4 + √(e³ - 4) and a = 4 - √(e³ - 4)Now, I need to determine the nature of these solutions. Let's compute √(e³ - 4). Since e³ is approximately 20.085, subtracting 4 gives 16.085. The square root of 16.085 is approximately 4.01. So, √(e³ - 4) ≈ 4.01.Therefore, the solutions are approximately:a ≈ 4 + 4.01 = 8.01a ≈ 4 - 4.01 = -0.01Wait a minute, but the original logarithmic function ln(a² - 8a + 20) requires that the argument a² - 8a + 20 is positive because the logarithm of a non-positive number is undefined. So, I need to check if these solutions make the argument positive.Let's check for a ≈ 8.01:a² - 8a + 20 ≈ (8.01)² - 8*(8.01) + 20 ≈ 64.1601 - 64.08 + 20 ≈ 20.0801, which is positive.For a ≈ -0.01:a² - 8a + 20 ≈ (-0.01)² - 8*(-0.01) + 20 ≈ 0.0001 + 0.08 + 20 ≈ 20.0801, which is also positive.So, both solutions are valid because they result in a positive argument for the logarithm.Now, looking back at the options:(A) two integers(B) one integer and one irrational number(C) two irrational numbers(D) two non-real numbers(E) no numbers, that is, the empty setI have two solutions, both of which are real numbers. But are they integers or irrational?From the approximate values, a ≈ 8.01 and a ≈ -0.01, neither of these are integers. Also, since √(e³ - 4) is an irrational number (because e is transcendental and e³ - 4 is not a perfect square), the solutions are irrational.Therefore, the nature of the solutions is two irrational numbers.Wait, but let me double-check. Since e³ is approximately 20.085, e³ - 4 is approximately 16.085, and the square root of that is approximately 4.01, which is not an integer. So, yes, both solutions are irrational.So, the correct answer should be (C) two irrational numbers.But hold on, let me think again. The quadratic equation I solved was a² - 8a + (20 - e³) = 0. The discriminant was -16 + 4e³, which is positive because e³ is about 20.085, so 4e³ is about 80.34, and 80.34 - 16 is 64.34, which is positive. So, the square root is real, meaning both solutions are real numbers.Since the discriminant is not a perfect square, the solutions are irrational. Therefore, both solutions are irrational numbers.Yes, that makes sense. So, the answer is (C) two irrational numbers.

💡Okay, so I have this problem about a hyperbola and a circle. Let me try to figure it out step by step. First, the hyperbola is given by the equation (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a > 0) and (b > 0). I know that for hyperbolas, the standard form is similar to this, and the foci are located at ((pm c, 0)), where (c^2 = a^2 + b^2). So, the right focus (F) should be at ((c, 0)).Next, there's a circle (C) with center (F), and its equation is given as (x^2 + y^2 - 4x + 3 = 0). I need to find the center and radius of this circle. To do that, I can rewrite the equation in standard form by completing the square.Starting with (x^2 + y^2 - 4x + 3 = 0), let me group the x-terms:[x^2 - 4x + y^2 + 3 = 0]Now, complete the square for the x-terms. The coefficient of x is -4, so half of that is -2, and squaring it gives 4. So, I add and subtract 4:[(x^2 - 4x + 4) + y^2 + 3 - 4 = 0]Simplify:[(x - 2)^2 + y^2 - 1 = 0]Which becomes:[(x - 2)^2 + y^2 = 1]So, the center of the circle is at ((2, 0)) and the radius is 1. Since the center of the circle is (F), that means the right focus of the hyperbola is at ((2, 0)). Therefore, (c = 2).From the hyperbola equation, we know that (c^2 = a^2 + b^2). Plugging in (c = 2), we get:[4 = a^2 + b^2]So, that's one equation relating (a) and (b).Now, the circle is tangent to the asymptotes of the hyperbola. I need to use this condition to find another equation involving (a) and (b). The asymptotes of the hyperbola (frac{x^2}{a^2} - frac{y^2}{b^2} = 1) are given by the equations:[y = pm frac{b}{a}x]These are straight lines passing through the origin with slopes (pm frac{b}{a}).Since the circle is tangent to these asymptotes, the distance from the center of the circle to each asymptote must be equal to the radius of the circle. The center of the circle is at ((2, 0)), and the radius is 1.The formula for the distance from a point ((x_0, y_0)) to the line (Ax + By + C = 0) is:[text{Distance} = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}]Let me rewrite the asymptote equations in the standard form (Ax + By + C = 0). Taking (y = frac{b}{a}x), we can rearrange it to:[frac{b}{a}x - y = 0]Similarly, the other asymptote is:[-frac{b}{a}x - y = 0]But since distance is always positive, I can just use one of them because the distance will be the same for both due to symmetry.So, using the asymptote (frac{b}{a}x - y = 0), the distance from the center ((2, 0)) to this line should be equal to the radius, which is 1.Plugging into the distance formula:[frac{|frac{b}{a} cdot 2 - 1 cdot 0 + 0|}{sqrt{left(frac{b}{a}right)^2 + (-1)^2}} = 1]Simplify the numerator:[left|frac{2b}{a}right| = frac{2b}{a}]Since (a) and (b) are positive, we can drop the absolute value.The denominator:[sqrt{left(frac{b^2}{a^2}right) + 1} = sqrt{frac{b^2 + a^2}{a^2}}} = frac{sqrt{a^2 + b^2}}{a}]But from earlier, we know that (a^2 + b^2 = 4), so:[sqrt{frac{4}{a^2}} = frac{2}{a}]Putting it all together, the distance is:[frac{frac{2b}{a}}{frac{2}{a}} = frac{2b}{a} cdot frac{a}{2} = b]So, the distance from the center to the asymptote is (b), and we know this distance must equal the radius of the circle, which is 1. Therefore:[b = 1]Now, going back to the earlier equation (a^2 + b^2 = 4), and substituting (b = 1):[a^2 + 1 = 4 implies a^2 = 3 implies a = sqrt{3}]So, we have (a = sqrt{3}) and (b = 1).Therefore, the standard equation of the hyperbola is:[frac{x^2}{3} - frac{y^2}{1} = 1]Which simplifies to:[frac{x^2}{3} - y^2 = 1]I think that's the answer. Let me just double-check my steps.1. Found the center and radius of the circle correctly by completing the square.2. Identified that the center is the focus, so (c = 2).3. Used the relationship (c^2 = a^2 + b^2) correctly.4. Wrote the asymptotes correctly and used the distance formula from the center to the asymptote.5. Calculated the distance correctly, simplified, and found (b = 1).6. Substituted back to find (a = sqrt{3}).7. Plugged into the standard hyperbola equation.Everything seems to check out. I don't see any mistakes in my reasoning.**Final Answer**The standard equation of the hyperbola is boxed{dfrac{x^2}{3} - y^2 = 1}.

💡Okay, so I have this problem where I need to find the value of the expression α(β - γ)² + β(γ - α)² + γ(α - β)², where α, β, and γ are the roots of the cubic polynomial x³ - 3x - 2 = 0. Hmm, that seems a bit complicated, but maybe I can break it down step by step.First, I remember that for a cubic polynomial like x³ + ax² + bx + c = 0, the sum of the roots, the sum of the products of the roots two at a time, and the product of the roots can be found using Vieta's formulas. In this case, the polynomial is x³ - 3x - 2, so comparing it to the general form, I see that the coefficient of x² is 0. That means:α + β + γ = 0 (since the coefficient of x² is 0)αβ + αγ + βγ = -3 (since the coefficient of x is -3)αβγ = 2 (since the constant term is -2, and Vieta's formula says it's -d/a, which here is -(-2)/1 = 2)Okay, so I have these three equations from Vieta's formulas. Now, I need to compute the expression α(β - γ)² + β(γ - α)² + γ(α - β)². Let me write that out:E = α(β - γ)² + β(γ - α)² + γ(α - β)²I think I should expand each term to see if I can simplify it. Let me start by expanding each squared term:(β - γ)² = β² - 2βγ + γ²(γ - α)² = γ² - 2γα + α²(α - β)² = α² - 2αβ + β²So, substituting these back into E:E = α(β² - 2βγ + γ²) + β(γ² - 2γα + α²) + γ(α² - 2αβ + β²)Now, let me distribute each α, β, and γ into their respective parentheses:E = αβ² - 2αβγ + αγ² + βγ² - 2βγα + βα² + γα² - 2γαβ + γβ²Hmm, let me rearrange the terms to group similar ones together:E = αβ² + αγ² + βγ² + βα² + γα² + γβ² - 2αβγ - 2βγα - 2γαβWait, I notice that the terms -2αβγ, -2βγα, and -2γαβ are all the same, just cyclic permutations. So, combining them:E = (αβ² + αγ² + βγ² + βα² + γα² + γβ²) - 6αβγNow, looking at the first part, αβ² + αγ² + βγ² + βα² + γα² + γβ², I can factor terms:Let me see, αβ² + βα² = αβ(α + β), similarly for the others.So, E can be written as:E = αβ(α + β) + βγ(β + γ) + γα(γ + α) - 6αβγBut from Vieta's formula, we know that α + β + γ = 0, so α + β = -γ, β + γ = -α, and γ + α = -β.Substituting these into the expression:E = αβ(-γ) + βγ(-α) + γα(-β) - 6αβγSimplify each term:E = -αβγ - βγα - γαβ - 6αβγWait, but each of these terms is the same: -αβγ, -βγα, -γαβ. So that's three terms of -αβγ:E = -3αβγ - 6αβγ = (-3 - 6)αβγ = -9αβγBut from Vieta's formula, αβγ = 2, so:E = -9 * 2 = -18Wait, that seems straightforward, but let me double-check my steps to make sure I didn't make a mistake.First, expanding the squares and distributing seems correct. Then, grouping the terms correctly, I noticed the -2αβγ terms adding up to -6αβγ. Then, for the other terms, I factored them as αβ(α + β) etc., which is correct because αβ² + βα² = αβ(α + β). Then, using α + β = -γ, etc., substituted correctly, leading to -αβγ each time. So three terms of -αβγ, which is -3αβγ. Then subtracting another 6αβγ gives -9αβγ. Since αβγ is 2, the result is -18.Hmm, that seems consistent. Alternatively, maybe I can approach this differently to verify.Another approach: Since α, β, γ are roots of x³ - 3x - 2 = 0, each satisfies x³ = 3x + 2. Maybe I can use that to simplify higher powers if needed.Looking back at E, after expanding, I had:E = αβ² + αγ² + βγ² + βα² + γα² + γβ² - 6αβγIs there another way to express αβ² + βγ² + γα² + βα² + γβ² + αγ²?Wait, that's equal to αβ² + βα² + βγ² + γβ² + γα² + αγ², which can be written as αβ(α + β) + βγ(β + γ) + γα(γ + α). As before, which leads to the same substitution.Alternatively, perhaps I can factor E differently or use symmetric sums.Wait, another thought: Let me consider that E is a symmetric function in α, β, γ. So maybe I can express it in terms of the elementary symmetric sums, which I already know from Vieta's formulas.Let me recall that in symmetric polynomials, expressions can often be written in terms of σ1 = α + β + γ, σ2 = αβ + αγ + βγ, and σ3 = αβγ.Given that σ1 = 0, σ2 = -3, σ3 = 2.Looking back at E, which is:E = αβ² + αγ² + βγ² + βα² + γα² + γβ² - 6αβγHmm, let me see if I can express αβ² + βγ² + γα² in terms of σ1, σ2, σ3.Wait, actually, I think there's a formula for the sum αβ² + βγ² + γα². Let me recall that:αβ² + βγ² + γα² = (α + β + γ)(αβ + βγ + γα) - 3αβγWait, is that correct? Let me check:Let me compute (α + β + γ)(αβ + βγ + γα):= α(αβ + βγ + γα) + β(αβ + βγ + γα) + γ(αβ + βγ + γα)= α²β + αβγ + αγ² + αβ² + β²γ + βγα + αβγ + βγ² + γ²αWait, that seems messy. Wait, perhaps a better way is to note that:αβ² + βγ² + γα² = (α + β + γ)(αβ + βγ + γα) - (α³ + β³ + γ³)Wait, let me see:(α + β + γ)(αβ + βγ + γα) = α²β + α²γ + β²α + β²γ + γ²α + γ²β + 3αβγSo, that's equal to (αβ² + αγ² + βα² + βγ² + γα² + γβ²) + 3αβγTherefore, αβ² + αγ² + βγ² + βα² + γα² + γβ² = (α + β + γ)(αβ + βγ + γα) - 3αβγBut since α + β + γ = 0, this simplifies to:= 0 * σ2 - 3αβγ = -3αβγSo, going back to E:E = (αβ² + αγ² + βγ² + βα² + γα² + γβ²) - 6αβγ= (-3αβγ) - 6αβγ= -9αβγWhich is the same as before. Since αβγ = 2, E = -9 * 2 = -18.So, both methods lead me to the same result, which gives me more confidence that the answer is indeed -18.Wait, but just to make sure, maybe I can compute E numerically by finding the actual roots of the polynomial and plugging them in. Let me try that.The polynomial is x³ - 3x - 2 = 0. Let me try to factor it. Maybe it has rational roots. By Rational Root Theorem, possible roots are ±1, ±2.Testing x=1: 1 - 3 - 2 = -4 ≠ 0x=-1: -1 + 3 - 2 = 0. Oh, so x=-1 is a root.So, we can factor (x + 1) out of the polynomial. Let's perform polynomial division or use synthetic division.Using synthetic division with root -1:Coefficients: 1 (x³), 0 (x²), -3 (x), -2 (constant)Bring down 1.Multiply by -1: 1*(-1) = -1. Add to next coefficient: 0 + (-1) = -1.Multiply by -1: -1*(-1) = 1. Add to next coefficient: -3 + 1 = -2.Multiply by -1: -2*(-1) = 2. Add to last coefficient: -2 + 2 = 0. Perfect.So, the polynomial factors as (x + 1)(x² - x - 2). Now, factor x² - x - 2: discriminant is 1 + 8 = 9, so roots are [1 ± 3]/2, which are 2 and -1.So, the roots are x = -1, x = 2, and x = -1 again? Wait, no, wait: x² - x - 2 factors as (x - 2)(x + 1), so the roots are x = 2 and x = -1. But wait, we already had x = -1, so the polynomial is (x + 1)^2 (x - 2). So, the roots are α = -1 (double root), β = -1, γ = 2.Wait, but in the problem statement, it's given as α, β, γ, so maybe they are considering all roots, including multiplicities. So, let's take α = -1, β = -1, γ = 2.Now, let's compute E = α(β - γ)² + β(γ - α)² + γ(α - β)².Plugging in the values:E = (-1)((-1) - 2)² + (-1)(2 - (-1))² + 2((-1) - (-1))²Compute each term:First term: (-1)((-1) - 2)² = (-1)(-3)² = (-1)(9) = -9Second term: (-1)(2 - (-1))² = (-1)(3)² = (-1)(9) = -9Third term: 2((-1) - (-1))² = 2(0)² = 2*0 = 0So, E = -9 + (-9) + 0 = -18That's the same result as before. So, this numerical approach confirms that E = -18.Wait, but hold on, in the problem statement, it's stated as "the roots", but in this case, the polynomial has a double root at x = -1 and a single root at x = 2. So, technically, the roots are -1, -1, and 2. So, when I plug them into E, I have to consider that two of them are the same.But in my calculation above, I considered α = -1, β = -1, γ = 2, and the expression evaluated to -18. So, that's consistent.Alternatively, if I had taken α = -1, β = 2, γ = -1, would the result be the same? Let me check:E = α(β - γ)² + β(γ - α)² + γ(α - β)²= (-1)(2 - (-1))² + 2((-1) - (-1))² + (-1)((-1) - 2)²Compute each term:First term: (-1)(3)² = (-1)(9) = -9Second term: 2(0)² = 0Third term: (-1)(-3)² = (-1)(9) = -9So, E = -9 + 0 + (-9) = -18Same result. So, regardless of the order, the value is -18.Therefore, I'm confident that the answer is -18.**Final Answer**boxed{-18}

💡Okay, so I've got this problem here: "There are 5 flowers and some bees. The difference of the number of flowers and bees over there is 2. How many bees are there?" Alright, let's try to figure this out step by step.First, I need to understand what the problem is asking. It mentions there are 5 flowers and an unknown number of bees. The key piece of information is that the difference between the number of flowers and bees is 2. So, I need to find out how many bees there are based on this difference.Let me break it down. The number of flowers is given as 5. Let's call the number of bees "B." The problem states that the difference between the number of flowers and bees is 2. Now, I need to interpret what "difference" means here. In math, the difference between two numbers is the result of subtracting one number from the other. So, if I subtract the number of bees from the number of flowers, the result is 2.Wait, but the problem doesn't specify which one is larger—whether there are more flowers than bees or more bees than flowers. That's something I need to consider. If the difference is 2, it could mean that there are 2 more flowers than bees or 2 more bees than flowers. So, I might have to consider both possibilities.Let's start by assuming that there are more flowers than bees. In that case, the number of flowers minus the number of bees equals 2. So, mathematically, that would be:5 - B = 2Now, I can solve for B. Subtracting 5 from both sides gives:-B = 2 - 5Which simplifies to:-B = -3Multiplying both sides by -1 to solve for B:B = 3So, if there are more flowers than bees, there would be 3 bees.But I should also consider the other possibility where there are more bees than flowers. In that case, the number of bees minus the number of flowers equals 2. So, the equation would be:B - 5 = 2Solving for B, I add 5 to both sides:B = 2 + 5Which gives:B = 7So, if there are more bees than flowers, there would be 7 bees.Now, the problem doesn't specify which scenario it is, so I need to see if there's any additional information or context that might help me decide between these two possibilities. The problem simply states that the difference is 2, without indicating which is larger. Therefore, both solutions are mathematically valid based on the information provided.However, in many real-world contexts, especially in problems like this, it's common to assume that the number being subtracted is smaller unless stated otherwise. So, if we take the first interpretation where there are more flowers than bees, the number of bees would be 3.But to be thorough, I should present both solutions and let the reader decide based on any additional context they might have. Since the problem doesn't provide further details, both answers are possible.In summary, depending on whether there are more flowers or more bees, the number of bees could be either 3 or 7. Without additional information, both solutions are valid.

💡First, I need to determine the time it takes for the car to travel 1 kilometer at the given speed of 48 km/h. To do this, I'll convert the speed from kilometers per hour to kilometers per second.Next, I'll calculate the time it takes to travel 1 kilometer at this speed. Then, I'll set up an equation that represents the relationship between the times at the two different speeds, incorporating the 15-second difference mentioned in the problem.After setting up the equation, I'll solve for the unknown speed ( v ) by isolating it on one side of the equation. This will involve basic algebraic manipulation to find the value of ( v ).Finally, I'll verify the solution to ensure that it makes sense within the context of the problem and that the calculations are accurate.