### Video Transcript

What is the surface-area-to-volume
ratio of the nanoparticle in this diagram.

In order to solve this problem, let
us think through what we will do. First, we will need to calculate
the surface area of the nanoparticle. Then, we will need to calculate the
volume of the nanoparticle. Finally, we can calculate the
surface-area-to-volume ratio of the nanoparticle.

Let us first start off by recalling
the surface area of a rectangular prism. This is the sum of all the areas of
each surface of all six sides. To calculate the area, we need to
multiply the width times the length of each side of the prism. Let us take a look at side 𝐴. This has an area of 25 nanometers
times 15 nanometers. Note that side 𝐵 is identical to
side 𝐴, so we can multiply this area by two, which is 750 nanometers squared. Now we only have four sides
left.

Next, let us identify side 𝐶 at
the top of the nanoparticle, which has an area of 50 nanometers times 25
nanometers. Side 𝐷 at the bottom of the
nanoparticle, which we cannot see, is identical to side 𝐶. So we can multiply this area by two
to have an area of 2500 nanometers squared. Now we only have two sides
remaining. The side facing towards us will be
labeled as side 𝐸. Note how side 𝐸 is identical to
side 𝐹 on the far side of the nanoparticle, which we cannot see.

The area of side 𝐸 is 15
nanometers times 50 nanometers. And we multiply this by two since
sides 𝐸 and 𝐹 are identical. This has a total area of 1500
nanometers squared. To calculate the total surface area
of the whole nanoparticle, we need to add all the areas together, which is 4750
nanometers squared.

Now since we have calculated the
surface area of the nanoparticle, let us calculate the volume of the nanoparticle,
which will be much quicker. Let us first discuss how to
calculate the volume of a rectangular prism, which is width times length times
height. The width, length, and height are
25 nanometers, 50 nanometers, and 15 nanometers, which is a total volume of 18750
nanometers cubed.

Finally, we can calculate the
surface-area-to-volume ratio of the nanoparticle by dividing 4750 nanometers squared
by 18750 nanometers cubed. It is important to place the
surface area first because that is what the problem is asking us to compare. In addition, it is important to
note the units of nanometers squared divided by nanometers cubed reduced to
nanometers to the negative first power. Therefore, the correct answer
choice is 0.25 nanometers to the negative first power.