### Video Transcript

A cylinder has a surface area of
256𝜋 square inches and a height of eight inches. Determine its volume in terms of
𝜋.

We begin by considering the
information given to us and what relevant formulas we can apply. We have been given the surface area
of a cylinder, which we recall is given by the formula 𝑆 equals two 𝜋 times 𝑟
times ℎ plus two 𝜋 times 𝑟 squared, where 𝑟 is the radius of the base and ℎ is
the height of the cylinder. We have also been asked to find the
volume, which is given by the formula 𝑉 equals 𝜋𝑟 squared ℎ. Since both formulas involve 𝑟, we
can relate the two. Specifically, we can use the
surface area formula to find the radius, which we can then substitute into the
volume formula to find 𝑉.

Now, we know the cylinder has a
height of eight inches. So we can substitute ℎ equals eight
into the surface area formula. Similarly, we also substitute 𝑆
equals 256𝜋 and simplify to get 256𝜋 equals 16𝜋𝑟 plus two 𝜋𝑟 squared. Next, we divide all terms on both
sides of the equation by the highest common factor, two 𝜋. Then, we get 128 equals eight 𝑟
plus 𝑟 squared.

We recognize this expression as a
quadratic trinomial. So we will rearrange the equation
into standard form and to equal zero, then try to solve by factoring. By subtracting 128 from each side
of the equation, we get zero equals 𝑟 squared plus eight 𝑟 minus 128. Since the leading term has a
coefficient of one, we start with 𝑟 in each parentheses.

We need to find two numbers with a
product of negative 128 and a sum of eight. Those two numbers are 16 and
negative eight. Thus, the factors of 𝑟 squared
plus eight 𝑟 minus 128 are 𝑟 plus 16 and 𝑟 minus eight. By setting each factor equal to
zero, we find that 𝑟 equals negative 16 and positive eight. Since the radius cannot have a
negative length, we determine that the radius is eight inches long.

Now that we have the radius and
height of the cylinder, we can find its volume by substituting 𝑟 equals eight and ℎ
equals eight. Keeping our answer in terms of 𝜋,
we find that the cylinder has a volume of 512𝜋 cubic inches.