Question

the radius of some planet is 1840 miles. use the formula for the radius r of a sphere given its surface area a,
$r = \sqrt{\frac{a}{4\pi}}$
to find the surface area of the planet.

$a = \square$ sq mi
(round to the nearest square mile as needed.)

Explanation:

Step1: Start with the given formula

We have the formula for the radius \( r \) of a sphere in terms of its surface area \( A \): \( r = \sqrt{\frac{A}{4\pi}} \). We need to solve for \( A \) when \( r = 1840 \) miles.

First, square both sides of the equation to eliminate the square root:
\( r^2=\frac{A}{4\pi} \)

Step2: Solve for \( A \)

Multiply both sides of the equation by \( 4\pi \) to isolate \( A \):
\( A = 4\pi r^2 \)

Step3: Substitute \( r = 1840 \) into the formula

Now substitute \( r = 1840 \) into the formula for \( A \):
\( A = 4\pi(1840)^2 \)

First, calculate \( (1840)^2 \):
\( 1840^2=1840\times1840 = 3385600 \)

Then, multiply by \( 4\pi \):
\( A = 4\pi\times3385600 \)
\( A = 13542400\pi \)

Now, calculate the numerical value. Using \( \pi\approx3.14159 \):
\( A\approx13542400\times3.14159 \)
\( A\approx13542400\times3 + 13542400\times0.14159 \)
\( A\approx40627200+191746.9816 \)
\( A\approx40818946.9816 \)

Rounding to the nearest square mile, we get \( A\approx40818947 \)

Answer:

\( 40818947 \)