Chapter 6.5, Problem 33WE

To determine

The observatory of a tall building in Chicago is 607 feet high then find the distance to horizon from this observatory if the distance $\left(d\right)$ in miles to the horizon is given for height $\left(h\right)$ as $d=\sqrt{\frac{3}{2}h}$ .

Answer to Problem 33WE

The distance $\left(d\right)$ is equal to 30.174 miles.

Explanation of Solution

Given information: The height of the building is 607 feet. And the distance $\left(d\right)$ in miles to the earth’s horizon for height $\left(h\right)$ is,

  $d=\sqrt{\frac{3}{2}h}\mathrm{....}\left(1\right)$

Calculation: Plugging for height $\left(h\right)$ equal to 607 feet in (1),

  $\begin{array}{c}d=\sqrt{\frac{3\left(607\right)}{2}}\mathrm{....}\left(\text{For }h=607\right)\\ =30.174\text{ miles}\end{array}$

To determine

Solve the formula for height $\left(h\right)$ from the relation $d=\sqrt{\frac{3}{2}h}$ .

Answer to Problem 33WE

The solution for height $\left(h\right)$ is $h=\frac{2d^2}{3}$ .

Explanation of Solution

Given information: The distance $\left(d\right)$ in miles to the earth’s horizon for height $\left(h\right)$ is,

  $d=\sqrt{\frac{3}{2}h}\mathrm{....}\left(2\right)$

Calculation: Squaring (2) and solving for height $\left(h\right)$ ,

  $\begin{array}{c}{d}^2=\frac{3}{2}h\mathrm{....}\left(\text{Squared}\right)\\ \text{or, }h=\frac{2d^2}{3}\end{array}$