Question

select the correct answer. the weight of an object, w, varies inversely as the square of its distance from the center of earth, d. when an astronaut stands in a training center on the surface of earth (3,960 miles from the center), she weighs 155 pounds. to the nearest tenth of a pound, what will be the approximate weight of the astronaut when she is standing on a space - station, in orbit 240 miles above the training center? a. 174.4 pounds b. 164.4 pounds c. 146.1 pounds d. 137.8 pounds

Explanation:

Step1: Write the inverse - square formula

The weight $w$ varies inversely as the square of the distance $d$ from the center of Earth, so $w=\frac{k}{d^{2}}$, where $k$ is a constant.
We know that when $d = 3960$ miles (at the surface of Earth), $w = 155$ pounds. Substitute these values into the formula to find $k$:
$155=\frac{k}{3960^{2}}$, then $k = 155\times3960^{2}$.

Step2: Calculate the distance when in orbit

The distance from the center of Earth when the astronaut is in orbit 240 miles above the training - center is $d=3960 + 240=4200$ miles.

Step3: Find the weight in orbit

Since $w=\frac{k}{d^{2}}$ and $k = 155\times3960^{2}$, substitute $d = 4200$ into the formula:
$w=\frac{155\times3960^{2}}{4200^{2}}$.
First, calculate $3960^{2}=3960\times3960 = 15681600$ and $4200^{2}=4200\times4200 = 17640000$.
Then $155\times3960^{2}=155\times15681600 = 2430648000$.
$w=\frac{2430648000}{17640000}=137.8$.

Answer:

D. 137.8 pounds