Chapter 8.2, Problem 27E

a.

To determine

after how many seconds does the water balloon hits the ground.

Answer to Problem 27E

3 Seconds

Explanation of Solution

Given:

Water balloon is dropped from a height of 144 feet.

Concept used:

The height (in feet) of a falling object t seconds after it is dropped from an initial height of $s_0$ (in feet) is given by the function:

  $f\left(t\right)=-16t^2+s_0$.......(1)

Calculation:

When the water balloon hits the ground its height is 0 feet.

Substituting $s_0=144$ and $f\left(t\right)=0$ in equation (1),

  $\begin{array}{c}f\left(t\right)=-16t^2+s_0\\ 0=-16t^2+144\\ 16t^2=144\end{array}$

  $\begin{array}{c}{t}^2=\frac{144}{16}\\ t^2=9\\ t=\pm 3\end{array}$

Now, here t represents time and time can never be negative, so solution is $t=3$ .

Conclusion:

The water balloon hits the ground, 3 seconds after it is dropped.

b.

To determine

the effect on part (a) if the initial height is adjusted by k feet.

Answer to Problem 27E

If $k>0$ the shift is k units up and so the balloon will take more than 3 seconds to hit the ground

And if $k<0$ the shift is $\left|k\right|$ units down and so the balloon will take less than 3 seconds to hit the ground

Explanation of Solution

Given:

Water balloon is dropped from a height of 144 feet.

Concept used:

The height (in feet) of a falling object t seconds after it is dropped from an initial height of $s_0$ (in feet) is given by the function:

  $f\left(t\right)=-16t^2+s_0$ ….. (1) Vertical translation:

The graph of $g\left(x\right)=f\left(x\right)+k$ is a vertical translation of the graph of f.

The vertical shift is k units up if $k>0$ and k units down if $k<0$ .

The function that gives the height of given water balloon is:

  $f\left(t\right)=-16t^2+144$

Now, if the initial height is adjusted by k feet, the new function will be

  $g\left(t\right)=-16t^2+144+k$

It can be observed that $g\left(t\right)=f\left(t\right)+k$

So, the graph of g is a vertical translation of the graph of f.

Now, if $k>0$ the shift is k units up and so the balloon will take more than 3 seconds to hit the ground

And if $k<0$ the shift is $\left|k\right|$ units down and so the balloon will take less than 3 seconds to hit the ground