### Video Transcript

Determine the height of a cylinder
given that its total surface area is 100 square centimeters and its radius equals
its height. Round your answer to two decimal
places.

We’ve been given the total surface
area of this cylinder, so let’s begin by recalling how this is calculated. The total surface area consists of
the areas of the circular ends of the cylinder along with the lateral area, which is
the area of the curved surface that wraps around the cylinder.

For a cylinder with a radius of 𝑟
units and a height of ℎ units, the formula for calculating the total surface area is
written as two 𝜋𝑟 squared plus two 𝜋𝑟ℎ. We know that the total surface area
of this cylinder is 100 square centimeters, so we can substitute this value into the
formula. We can’t yet solve this equation
though because it contains two unknowns, 𝑟 and ℎ.

However, we’re told that the radius
of this cylinder is equal to its height. This means we can replace either ℎ
or 𝑟 with the other to give an equation in one variable only. As it’s the height we’ve been asked
to calculate, we’ll choose to replace both the 𝑟’s with ℎ’s so that we have an
equation in ℎ only. Doing so gives the equation 100
equals two 𝜋ℎ squared plus two 𝜋ℎ squared.

Simplifying by combining the terms
on the right-hand side gives 100 equals four 𝜋ℎ squared. We now solve this equation for ℎ by
first dividing both sides by four 𝜋. That gives ℎ squared equals 100
over four 𝜋. And simplifying gives ℎ squared
equals 25 over 𝜋. The final step is to take the
square root of each side of the equation, taking only the positive solution here as
ℎ represents a length and so is by definition positive.

We have that ℎ is equal to the
square root of 25 over 𝜋. In simplified form, that’s five
over root 𝜋. And as a decimal, it’s equal to
2.8209 continuing. We’re asked to give the answer to
two decimal places. So rounding as required and
including the length units, which are centimeters, we have that the height of this
cylinder and also the length of its radius is 2.82 centimeters to two decimal
places.