Hi, I can't figure this story problem out can u please help? I need to learn how to do this story problem.
You dive directly upward on a board that is 32 feet high. After t seconds, your hight above the water is described by the polynominal -16t^2+16t+32.
a) factor the polynominal completely. Begin by factoring -16 from each term.
b) Evaluate both the origional and its factored form for t=2. Do you get the same answer? Describe what this answer means.
Of course, I'd be happy to help you with this story problem! To solve this problem, we'll first start by factoring the given polynomial, which will give us its factored form. Then, we'll evaluate both the original and the factored forms for t=2 and compare the results.
a) Factoring the polynomial:
The given polynomial is -16t^2 + 16t + 32. Let's begin by factoring out -16 from each term:
-16t^2 + 16t + 32 = -16(t^2 - t - 2)
Now, we have (t^2 - t - 2) remaining inside the parentheses. To factor this quadratic expression further, we need to find two numbers whose product is -2 and whose sum is -1 (since the coefficient of t is -1). These numbers are -2 and +1. So, we can factor the quadratic expression as follows:
(t^2 - t - 2) = (t - 2)(t + 1)
Hence, the factored form of the polynomial is -16(t - 2)(t + 1).
b) Evaluating the original and factored forms for t=2:
Now, let's evaluate both the original and factored forms of the polynomial for t=2.
Original form: -16t^2 + 16t + 32
Substituting t=2, we have:
-16(2)^2 + 16(2) + 32 = -16(4) + 32 + 32 = -64 + 32 + 32 = 0
Factored form: -16(t - 2)(t + 1)
Substituting t=2:
-16(2 - 2)(2 + 1) = -16(0)(3) = 0
As you can see, both the original and factored forms evaluated for t=2 yield a result of 0. This means that your height above the water is 0 feet at t=2 seconds. In other words, at exactly 2 seconds, you are at the water surface.
I hope this explanation helps you understand how to approach this story problem. If you have any further questions, feel free to ask!