Chapter 7.3, Problem 30PPS

a.

To determine

An equation that models an equation $V=\sqrt{v^2+64h}$ where $\left(v\right)$ is the initial velocity in feet per second, $\left(V\right)$ is the velocity of the roller when it moves down the hill and $\left(h\right)$ is vertical drop in feet.

Answer to Problem 30PPS

The equation $\sqrt{100+64h}$ models the situation $V=\sqrt{v^2+64h}$ .

Explanation of Solution

Given information: The relation between the velocity of roller $\left(V\right)$ , velocity of coaster $\left(v\right)$ and height $\left(h\right)$ of the hill is given as,

  $V=\sqrt{v^2+64h}\mathrm{......}\left(1\right)$

According to question $v=10$ feet per second is the initial velocity of the coaster at the top of the hill. And velocity of roller $\left(V\right)$ is 90 feet per second.

Calculation: Plugging for $v=10$ in (1) and simplify for distance $\left(V\right)$ .Therefore from (1),

  $\begin{array}{c}V=\sqrt{{\left(10\right)}^2+64h}\mathrm{......}\left(\text{For }v=10\right)\\ =\sqrt{100+64h}\mathrm{......}\left(2\right)\end{array}$

b.

To determine

The height of the hill that the designer will make.

Answer to Problem 30PPS

The height of the hill will be $125\text{feet}$ .

Explanation of Solution

Given information: According to question $v=10$ feet per second is the initial velocity of the coaster at the top of the hill. And when the roller moves down the hill its velocity $\left(V\right)$ is 90 feet per second.

Calculation: Plugging for $V=90$ in (1) and simplify for height $\left(h\right)$ .Therefore from (1),

  $\begin{array}{c}\left(90\right)=\sqrt{100+64h}\mathrm{......}\left(\text{For }V=10\right)\\ 8100=100+64h\mathrm{......}\left(\text{Squared}\right)\\ \text{or, }64h=8000\\ h=125\text{feet}\end{array}$