Chang is measuring the speeds of stars as they travel around a black hole. She notices that the speed, s, in kilometers per second t days after March 1^("st ") is given by: s(t)=100+30 t-(t-2)^(2) Which of the following expressions for the star's speed is equivalent to the given expression and contains the maximum speed of the star as a constant or coefficient? Choose 1 answer: (A) 96+34 t-t^(2) (B) -(t-17)^(2)+385 (c) -(t-34)^(2)+385 (D) 104+30 t-t^(2)

Question

Chang is measuring the speeds of stars as they travel around a black hole. She notices that the speed,  s, in kilometers per second  t days after March  1^("st ") is given by:  s(t)=100+30 t-(t-2)^(2) Which of the following expressions for the star's speed is equivalent to the given expression and contains the maximum speed of the star as a constant or coefficient? Choose 1 answer: (A)  96+34 t-t^(2) (B)  -(t-17)^(2)+385 (c)  -(t-34)^(2)+385 (D)  104+30 t-t^(2)

Bytelearn · Accepted Answer

Given expression: We are given the expression for the star's speed as a function of time: s(t) = 100 + 30t - (t - 2)² To find an equivalent expression that contains the maximum speed of the star as a constant or coefficient, we need to complete the square for the quadratic part of the expression. First, let's expand the quadratic term: (t - 2)² = t² - 4t + 4 Now, substitute this back into the original expression: s(t) = 100 + 30t - (t² - 4t + 4)Expand quadratic term: Combine like terms in the expression: s(t) = 100 + 30t - t² + 4t - 4 s(t) = 96 + 34t - t² This is the expanded form of the star's speed function.Combine like terms: To complete the square, we need to find the value that makes the quadratic term a perfect square trinomial. The coefficient of t in the quadratic term is -1, so we take half of the linear coefficient 34, which is 17, and square it to get 289. Now we add and subtract 289 inside the function to complete the square: s(t) = 96 + 34t - t² + 289 - 289 s(t) = 96 - 289 + 34t - (t² - 34t + 289) s(t) = -193 + 34t - (t - 17)²Complete the square: We made a mistake in the previous step by subtracting 289 instead of adding it to both sides. Let's correct this: s(t) = 96 + 34t - t² + 289 - 289 s(t) = 96 + 289 + 34t - t² - 34t + 289 s(t) = 385 + 34t - t² - 34t + 289 s(t) = 385 - (t² - 34t + 289) s(t) = 385 - (t - 17)² This is the correct form of the expression that contains the maximum speed of the star as a constant or coefficient.

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