Question

zach is standing on a hill 80 feet high. he throws a baseball upward with an initial velocity of 64 feet per second. the height of the ball h(t) in terms of the time t since the ball was thrown can be found using the function $h(t) = -16t^2 + 64t + 80$. how long does it take the ball to hit the ground? 2 seconds 3 seconds -1 seconds 5 seconds

Explanation:

Step1: Set height to 0

When the ball hits the ground, \( h(t) = 0 \). So we solve the equation \( -16t^2 + 64t + 80 = 0 \). Divide the entire equation by -16 to simplify: \( t^2 - 4t - 5 = 0 \).

Step2: Factor the quadratic

Factor \( t^2 - 4t - 5 \). We need two numbers that multiply to -5 and add to -4. Those numbers are -5 and 1. So the factored form is \( (t - 5)(t + 1) = 0 \).

Step3: Solve for t

Set each factor equal to zero: \( t - 5 = 0 \) or \( t + 1 = 0 \). This gives \( t = 5 \) or \( t = -1 \). Since time cannot be negative, we discard \( t = -1 \).

Answer:

5 seconds (corresponding to the option "5 seconds")