Question

1. a football is kicked from the ground with an initial vertical velocity of 80 feet per second. its height above the ground is modeled by the function $h(t) = -16t^2 + 80t$, where $t$ is the time in seconds since the football was kicked. how many seconds will it take the football to reach a height of 96 feet as it travels upward?

Explanation:

Step1: Set up the equation

We know the height function \( h(t)= - 16t^{2}+80t \), and we want to find \( t \) when \( h(t) = 96 \). So we set up the equation:
\( -16t^{2}+80t=96 \)

Step2: Rearrange the equation to standard quadratic form

Subtract 96 from both sides of the equation:
\( - 16t^{2}+80t - 96=0 \)
We can simplify this equation by dividing all terms by - 16:
\( t^{2}-5t + 6=0 \)

Step3: Factor the quadratic equation

We need to find two numbers that multiply to 6 and add up to - 5. The numbers are - 2 and - 3. So the factored form is:
\( (t - 2)(t - 3)=0 \)

Step4: Solve for t

Using the zero - product property, if \( (t - 2)(t - 3)=0 \), then \( t - 2 = 0 \) or \( t - 3=0 \). So \( t = 2 \) or \( t = 3 \).

Since the football is traveling upward, we take the smaller value of \( t \) (because when \( t = 3 \), the football is on its way down).

Answer:

2