### Video Transcript

The volume π of a sphere of radius
π is given by the formula π is equal to four-thirds times π times π cubed. The surface area π΄ of the same
sphere is given by π΄ equals four times π times π squared. Which of the following equations
correctly gives the ratio of the volume of the sphere to the surface area of the
sphere, π divided by π΄? (A) π divided by π΄ is equal to
one-third π cubed. (B) π divided by π΄ is equal to
one-third times π times π. (C) π divided by π΄ is equal to
one-fourth times π times π squared. (D) π divided by π΄ is equal to
one-third times π. (E) π divided by π΄ is equal to
one-third times π squared.

OKay, so in this exercise, weβre
considering a sphere that has a radius π. Weβre told the equations for the
volume as well as the surface area of the sphere in terms of its radius. And we want to solve for the ratio
of its volume to that surface area. We can do this by writing out the
expression for these two terms and then dividing the volume equation by the area
one. When we do this, weβre effectively
creating one equation out of two fractions. And notice that the fraction on the
left is exactly the ratio we want to solve for.

So then, looking on the right-hand
side of this equality, letβs see what will cancel out. Thereβs a factor of four in both
numerator and denominator, also a factor of π. And moreover, we can see that two
factors of the radius π cancel from top and bottom. When we remove all the cancelled
factors, weβre left with π, the radius of the sphere, divided by three. Looking across our five answer
options, we see that option (D) expresses the same ratio. The volume of a sphere to its
surface area is equal to one-third times its radius.