Video Transcript
Find, to the nearest tenth, the
volume of a sphere given that the area of its great circle is 400𝜋 square
inches.
Let’s begin by sketching a sphere
and recalling what we mean by its great circle. When a sphere is cut by a plane
that passes through its center, the intersection of the sphere and the plane is
called a great circle. A great circle cuts a sphere
exactly in half, and its radius is the same as that of the sphere itself.
We’re given that the area of the
great circle of a sphere is 400𝜋 square inches. And we’re asked to find the volume
of the sphere. We can use the fact that the area
of a circle is 𝜋𝑟 squared, where 𝑟 is the radius of the circle, to find the
radius of the sphere. And we do this by equating the area
of our great circle, that’s 400𝜋, with 𝜋𝑟 squared. Next, dividing through by 𝜋, we
have 𝑟 squared equal to 400. And then taking the positive square
root on both sides, positive since 𝑟 is a length and lengths are always positive,
we have 𝑟 equal to 20 inches.
So now that we have the radius of
the great circle, which equals the radius of the sphere, recalling that the volume
of a sphere is given by 𝑉 equals four over three 𝜋𝑟 cubed and making a little
space for our working out, we have 𝑉 equals four over three 𝜋 times 20 cubed. That’s four over three 𝜋 times
8000, which evaluates to 33,510.3216 and so on. To the nearest tenth, that’s to one
decimal place, this is 33,510.3.
Hence, given that the area of the
great circle of a sphere is 400𝜋 square inches, its volume is 33,510.3 cubic
inches, to the nearest tenth.
