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 = \boxed{} sq mi
(round to the nearest square mile as needed.)

Explanation:

Step1: Recall the formula for the surface area of a sphere

The formula for the radius \( r \) of a sphere in terms of its surface area \( A \) is \( r = \sqrt{\frac{A}{4\pi}} \). We need to solve for \( A \). First, square both sides of the equation to get rid of the square root: \( r^2=\frac{A}{4\pi} \).

Step2: Solve for \( A \)

Multiply both sides of the equation \( r^2 = \frac{A}{4\pi} \) by \( 4\pi \) to isolate \( A \). So, \( A = 4\pi r^2 \).

Step3: Substitute the given radius value

We know that \( r = 1840 \) miles. Substitute \( r = 1840 \) into the formula \( A = 4\pi r^2 \). First, calculate \( r^2 \): \( r^2=(1840)^2 = 1840\times1840 = 3385600 \). Then, multiply by \( 4\pi \): \( A = 4\pi\times3385600 \). Calculate \( 4\times3385600 = 13542400 \), so \( A = 13542400\pi \).

Step4: Calculate the numerical value

Now, calculate the value of \( 13542400\pi \). Using \( \pi\approx3.14159 \), we have \( A\approx13542400\times3.14159 \). Calculate \( 13542400\times3.14159 \approx 13542400\times3 + 13542400\times0.14159 \). \( 13542400\times3 = 40627200 \), and \( 13542400\times0.14159\approx13542400\times0.1 + 13542400\times0.04 + 13542400\times0.00159 \). \( 13542400\times0.1 = 1354240 \), \( 13542400\times0.04 = 541696 \), \( 13542400\times0.00159\approx21532.416 \). Adding these together: \( 1354240+541696 = 1895936 \), then \( 1895936 + 21532.416 = 1917468.416 \). Then add to \( 40627200 \): \( 40627200+1917468.416 = 42544668.416 \). Alternatively, using a calculator directly for \( 13542400\pi \approx13542400\times3.1415926535\approx42544667.6 \). Rounding to the nearest square mile, we get \( A\approx42544668 \) square miles.

Answer:

\( 42544668 \) (rounded to the nearest square mile)