Video Transcript
Find, to the nearest tenth, the
surface area of a sphere whose great circle has a circumference of 140𝜋 feet.
In this example, we’re given the
circumference of the great circle of a sphere, and we want to find the surface area
of the sphere. Now, we know that a great circle is
the largest circle that can be drawn on any given sphere and that as such, it cuts
the sphere in half and has the same radius as the sphere. We also know that the circumference
of a circle with radius 𝑟 is given by two 𝜋 times 𝑟. And we can use the circumference of
the great circle we’ve been given, that’s 140𝜋, to find the value of 𝑟 for both
the great circle and the sphere. We’ll then be able to use this
value for 𝑟 to calculate the surface area of the sphere.
Substituting the given
circumference into the formula, we have 140𝜋 equals two 𝜋𝑟. And now dividing both sides by two
𝜋, we have 𝑟 equal to 70. And now that we have the radius of
the great circle and hence the sphere, we can use this to find the surface area of
the sphere, where the surface area of a given sphere of radius 𝑟 is four 𝜋 times
𝑟 squared. Substituting 𝑟 equal to 70 into
this then, we have the surface area of the sphere equal to four 𝜋 times 70 squared,
which on squaring 70 is four 𝜋 times 4900. Evaluating this gives us a surface
area of 61575.216 and so on.
We’re asked to give our answer to
the nearest tenth, which is the first place after the decimal point and is a
two. The digit following this is a one,
so we can just round down to two. Since the circumference was given
in feet, our surface area will be in units of square feet. Hence, to the nearest tenth, the
surface area of the sphere is 61575.2 square feet.
