Video Transcript
The height of a cylinder is equal
to its base radius, and the surface area is 72𝜋 square centimeters. Find the height of the
cylinder.
We’ve been given the surface area
of this cylinder, so let’s begin by recalling how this is calculated. For a cylinder with a radius of 𝑟
units and a height of ℎ units, the total surface area is equal to two 𝜋𝑟 squared
plus two 𝜋𝑟ℎ. The two 𝜋𝑟 squared comes from the
areas of the circular ends of the cylinder, and two 𝜋𝑟ℎ is the area of the curved
surface that wraps around it. We know that the total surface area
of this cylinder is 72𝜋 square centimeters, so we can substitute this value on the
left-hand side.
We can’t solve this equation
though, because it contains two unknowns, 𝑟 and ℎ. However, we’re told that the height
of the cylinder is equal to its base radius. This means we can substitute either
𝑟 or ℎ for the other, giving an equation in one variable only. As it is the height of the cylinder
we’ve been asked to calculate, we’ll substitute ℎ in place of 𝑟 to give an equation
in ℎ only. Doing so gives 72𝜋 equals two 𝜋ℎ
squared plus two 𝜋ℎ squared. We can simplify this equation by
combining the terms on the right-hand side to give 72𝜋 equals four 𝜋ℎ squared.
We can then divide both sides of
the equation by four 𝜋 to give 18 equals ℎ squared. We solve this equation for ℎ by
square rooting, taking only the positive value as ℎ represents a length and so must
be positive. We have then that ℎ is equal to the
square root of 18. We want to simplify this radical if
possible, which we can do by recalling that 18 is equal to nine multiplied by two,
and nine is a square number. Hence, we have that ℎ is equal to
the square root of nine times two, which simplifies to three root two. Including the length units, we’ve
found that the height of this cylinder, which is also the length of its radius, is
three root two centimeters.
