Q 6.110 - Intermediate Algebra

Question

Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from$128$ feet above the ground, the function $h\left(t\right) = -16t^2 + 32t + 128$ models the height, h, of the penny above the ocean as a function of time, t. Find:

(a) the zeros of this function which is when the penny will hit the ocean

(b) when the penny will be $128$ feet above the ocean.

(c) the height the penny will be at $t=1$seconds which is when the penny will be at its highest point.

Step-by-Step Solution

Answer

Part a. The penny will hit the ocean $4$seconds after it is thrown.

Part b. The penny will be$128$ feet above the ocean after $2$ seconds.

Part c. The height of the penny at $t=1$ seconds is $144$ feet which are when the penny will be at its highest point.

Part a Step 1. Given Information

The given function is

$h\left(t\right) = -16t^2 + 32t + 128$

We have to find the zeros of this function which is when the penny will hit the ocean.

Part a Step 2. Finding the zeros of the function

To find the zeros of the function,

Put $h\left(t\right)=0$

$$\begin{gathered} -16t^2 + 32t + 128 = 0 \\ -16(t^2 - 2t - 8)=0 \\ -16(t+2)(t-4)=0 \end{gathered}$$

Part a Step 3. Use the Zero Product Property

The equation we get is

$-16\left[\right(t+2\left)\right(t-4\left)\right]=0$

By using the zero product property 

$-16 \ne 0$ , $$\begin{gathered} t+2=0 \\ t=-2 \end{gathered}$$ , $$\begin{gathered} t-4=0 \\ t=4 \end{gathered}$$

The $t=4$ tells us the penny will hit the ocean $4$ seconds after it is thrown. Since time cannot be negative, the result $t=-2$ is discarded.

Part b Step 1. Given Information

Calib throws the penny upward from $128$ feet above the ground, the function $h\left(t\right) = -16t^2 + 32t + 128$models the height, h, of the penny above the ocean as a function of time, t.

We have to find when the penny will be $128$ feet above the ocean.

Part b Step 2. To find when the penny will be 128 feet above the ocean

Put $h\left(t\right)=128$$h\left(t\right)=128$

Now, substitute the function 

$$\begin{gathered} -16t^2+ 32t + 128 = 128 \\ -16t^2 +32t =0 \\ 32t = 16t^2 \\ 2= t \end{gathered}$$

When Calib throws the penny, the penny will be $128$ feet above the ocean after $2$ seconds.  

Part c Step 1. Given Information

Calib throws the penny upward from $128$feet above the ground, the function $h\left(t\right)=-16t^2+32t+128$  models the height, h, of the penny above the ocean as a function of time, t.

We have to find the height the penny will be at $t=1$ second which is when the penny will be at its highest point.

Part c Step 2. To find the height of the penny at t = 1 seconds

Substitute$1$ for t in h(t)

$$\begin{gathered} h\left(1\right)=-16{\left(1\right)}^2+32\left(1\right)+128 \\ h\left(1\right)=-16+32+128 \\ h\left(1\right)=144 \end{gathered}$$

After $1$ second the penny will be at $144 feet.$