Q. 444 - Intermediate Algebra

Question

Solve:

Shruti is going to throw a ball from the top of a cliff. When she throws the ball from $80 feet$ above the ground, the function $h\left(t\right)=-16t^2+64t+80$ models the height $h$, of the ball above the ground as a function of time $t$. Find:

(a) The zeros of this function which tell us when the ball will hit the ground.

(b) The time(s) the ball will be $80 feet$ above the ground.

(c) The height of the ball will be at $t=2 seconds$which is when the ball will be at its highest point.

Step-by-Step Solution

Answer

(a) The zeros which tell us when the ball will hit the ground is $t=1 second$.

(b) The ball will be $80 feet$ above the ground when $t=4 seconds$.

(c) The height of the ball at $t=2 seconds$ is $144 feet$.

Step 1. Given the information.

The function given is,

$h\left(t\right)=-16t^2+64t+80$.

Part (a) Step 1. Finding the zeros.

Setting $h\left(t\right)=0$,

$-16t^2+64t+80=0$

Taking out $16$ common we get,

$$\begin{gathered} 16(-t^2+4t+5)=0 \\ -t^2+4t+5=0 \end{gathered}$$

Splitting the middle term such that the sum of the terms is $4$ and the product is $(-5)$.

$$\begin{gathered} -t^2+5t-t+5=0 \\ (-t^2+5t)+(-t+5)=0 \end{gathered}$$

Taking out the common factors,

$$\begin{gathered} t(5-t)+1(5-t)=0 \\ (t+1)(5-t)=0 \\ t=5 \\ or, t=-1 \end{gathered}$$

Since time cannot be negative.

The time will be $t=5$.

Part (b) Step 1. Finding the time when height is 80   f e e t .

Assuming the height be, $h\left(t\right)=80$

$$\begin{gathered} -16t^2+64t+80=80 \\ -16t^2+64t=0 \end{gathered}$$

Taking out $4$ common we get,

$$\begin{gathered} 16t(-t+4)=0 \\ -t+4=0 \\ t=4 \end{gathered}$$

The ball will be above the ground when $t=4 seconds.$

Part (c). Step 1. Finding the height when the time is t = 2   s e c o n d s .

The function is,

$h\left(t\right)=-16t^2+64t+80$

Substituting $t=2$,

$$\begin{gathered} h\left(2\right)=-16{\left(2\right)}^2+64\left(2\right)+80 \\ =-64+128+80 \\ =144 \end{gathered}$$

The ball will be at $144 feet$ when the time is, $t=2$.