Region About The North Pole

Calculus Level 3

Pretend that Earth is a perfectly spherical object with a radius of 4,000 miles. (In reality, Earth is not perfectly spherical, and its average radius is about 1% less than that.)

What is the area of the region within $\frac{4000\pi}{3}$ miles from the North Pole?

8000000\pi

16000000\pi

\frac{16000000\pi^3}{9}

2000000\pi^3

\frac{8000000\pi^3}{3}

1 solution

Henry Maltby
May 18, 2016

Relevant wiki: Spherical Geometry

The sphere has radius $R = 4000$ . The region in question is a circle on the sphere, whose center is in the interior of the sphere. The angle between the spherical radius pointing to the North Pole and a spherical radius pointing to the boundary of the circle is $\tfrac{1}{4000} \cdot \tfrac{4000\pi}{3} = \tfrac{\pi}{3}$ . Thus, the radius of the circle is $r = 4000 \sin(\tfrac{\pi}{3}) = 2000 \sqrt{3}$ . It follows that the area of the circle on the sphere is

$2\pi R \cdot (R - \sqrt{R^2 - r^2}) = 2\pi \cdot 4000 \cdot 2000 = \boxed{16000000\pi.}$

This is also one-third (the ratio of the angle $\tfrac{\pi}{3}$ to $\pi$ ) of the total surface area of the sphere.

Well, I did get that answer, but it seems that the solutions are wrong. I know this is an old question, but mind as well check it out.

Francisco Nascimento - 3 years, 2 months ago

Also, this is only one-quarter of the total surface area of the sphere, not one third.

Anthony Lamanna - 1 year, 6 months ago

Then, the solution is wrong!!!!

Paul Romero - 6 months, 3 weeks ago

Region About The North Pole

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