Video Transcript
If the total surface area of a
rectangular prism with a square base is 40 square centimeters and its height is
twice its width, find the total surface area of a cylinder inscribed inside the
rectangular prism.
We’re asked to find the surface
area of a cylinder inscribed in a rectangular prism, so let’s first consider what
this means. To say that a cylinder is inscribed
in a rectangular prism means that the cylinder touches every internal side of the
prism. And this means that the diameter of
the cylinder is the same as the width of the prism and also that they have the same
height.
Now, we’re told that the height of
the rectangular prism is twice its width. So, if we call the width 𝑥, then
the diameter of the cylinder is also equal to 𝑥 and the height of both the cylinder
and the rectangular prism equals two 𝑥. We’re also told that the surface
area of the rectangular prism is 40 square centimeters.
Now we recall that since our prism
has a square base, its surface area is given by two 𝑤 squared plus four 𝑤ℎ, where
𝑤 is its width and ℎ its height. In our case, this is two 𝑥 squared
plus four times 𝑥 times two 𝑥, since the width is 𝑥 and the height is two 𝑥. The surface area of the prism is
then equal to two 𝑥 squared plus eight 𝑥 squared, which is 10𝑥 squared.
Now we know that the surface area
of the rectangular prism is 40 square centimeters. So, leaving out the units for the
moment, we have 10𝑥 squared equals 40. And dividing both sides by 10 gives
𝑥 squared equal to four. Now taking the positive square root
on both sides, since 𝑥 is a length and therefore positive, we have 𝑥 equals
two. Hence, the width 𝑥, and therefore
the diameter of the cylinder, is equal to two centimeters. Now, making some space, we know
that the diameter of any circle is twice its radius. Hence, the radius of our cylinder
must equal one centimeter.
Next, we recall that the surface
area of a cylinder is two 𝜋𝑟ℎ plus two 𝜋𝑟 squared, where 𝑟 is its radius and ℎ
its height. And with 𝑟 equal to one centimeter
and the height equal to twice the diameter, that is, two 𝑥, which is four
centimeters, we have the surface area of the cylinder equals two 𝜋 times one times
four plus two 𝜋 times one squared. This evaluates to eight 𝜋 plus two
𝜋, which is equal to 10𝜋.
And so, the surface area of the
cylinder inscribed in the rectangular prism is 10𝜋 square centimeters.
