Chapter 8.1, Problem 78AYU

Surveillance Satellites A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

(a) Find an equation that relates the central angle $\theta$ to the height A.

(b) Find an equation that relates the observable distance $d$ and $\theta$.

(c) Find an equation that relates $d$ and $h$.

(d) If $d$ is to be 2500 miles, how high must the satellite orbit above Earth?

(e) If the satellite orbits at a height of 300 miles, what distance $d$ on the surface can be observed?

To determine

a. Find an equation that relates the central angle $\text{θ}$ to the height h.

Answer to Problem 78AYU

a. $\cos \left(\frac{\theta }{2}\right)=\frac{3960}{3960+h}$

Explanation of Solution

Given:

A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

Formula used:

$\text{cos}A=\frac{\text{adj side}}{\text{Hyp}}$

Calculation:

a. $\cos \left(\frac{\theta }{2}\right)=\frac{3960}{3960+h}$

To determine

b. Find an equation that relates the observable distance $d$ and $\text{θ}$.

Answer to Problem 78AYU

b. $d=3960\theta$

Explanation of Solution

Given:

A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

Formula used:

$\text{cos}A=\frac{\text{adj side}}{\text{Hyp}}$

Calculation:

b. $d=r\theta$

To determine

c. Find an equation that relates $d$ and $h$.

Answer to Problem 78AYU

c. $\cos \left(\frac{d}{7920}\right)=\frac{3960}{3960+h}$

Explanation of Solution

Given:

A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

Formula used:

$\text{cos}A=\frac{\text{adj side}}{\text{Hyp}}$

Calculation:

c. From a. and b, $\cos \left(\frac{d}{7920}\right)=\frac{3960}{3960+h}$.

To determine

d. If $d$ is to be 2500 miles, how high must the satellite orbit above Earth?

Answer to Problem 78AYU

d. $h\approx 206$ miles.

Explanation of Solution

Given:

A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

Formula used:

$\text{cos}A=\frac{\text{adj side}}{\text{Hyp}}$

Calculation:

d. $d=2500$ miles.

$\cos \left(\frac{2500}{7920}\right)=\frac{3960}{3960+h}$

$0.9506\approx \frac{3960}{3960+h}$

$3764+0.9506h\approx 3960$

$h\approx 206$ miles.

To determine

e. If the satellite orbits at a height of 300 miles, what distance $d$ on the surface can be observed?

Answer to Problem 78AYU

e. $d\approx 2990$ miles.

Explanation of Solution

Given:

A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration on the following page.

Formula used:

$\text{cos}A=\frac{\text{adj side}}{\text{Hyp}}$

Calculation:

e. $\cos \left(\frac{d}{7920}\right)=\frac{3960}{4260}$

$d=7920 {\cos }^{-1}\left(\frac{3960}{4260}\right)$

$d\approx 7920*0.378$ radians.

$d\approx 2990$ miles.