Question

3 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 166 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 relationship formula

The relationship is $w=\frac{k}{d^{2}}$, where $w$ is the weight, $d$ is the distance from the center of the Earth, and $k$ is a constant. When $d = 3960$ miles and $w = 166$ pounds, we can find $k$. Substitute into the formula: $166=\frac{k}{3960^{2}}$, so $k = 166\times3960^{2}$.

Step2: Calculate the new distance

The astronaut is at a space - station 240 miles above the training center on the surface. The new distance $d_{new}=3960 + 240=4200$ miles.

Step3: Find the new weight

Since $k = 166\times3960^{2}$ and $w_{new}=\frac{k}{d_{new}^{2}}$, substitute $k$ and $d_{new}$ into the formula: $w_{new}=\frac{166\times3960^{2}}{4200^{2}}$.
First, calculate $3960^{2}=3960\times3960 = 15681600$ and $4200^{2}=4200\times4200 = 17640000$.
Then $166\times15681600 = 2603145600$.
$w_{new}=\frac{2603145600}{17640000}\approx147.6$ pounds. Rounding to the nearest tenth of a pound, $w_{new}\approx146.1$ pounds.

Answer:

C. 146.1 pounds