Video Transcript
The graph shows how the volume 𝑉,
the surface area 𝐴, and the volume to surface area ratio, 𝑉 divided by 𝐴, vary
with the radius of a sphere. Which line represents 𝑉? Which line represents 𝐴? Which line represents 𝑉 divided by
𝐴?
Looking at this graph, we see that
both the vertical and the horizontal axes are unlabeled, but that we have these
three curves, the red one, the blue one, and the purple one. We’re told that one of these three
curves represents how the volume of a sphere varies with its radius. Another one represents how its
surface area changes with its radius. And then the third line shows how
the ratio of volume to surface area changes with sphere radius. We want to figure out which line is
which.
We notice that all three lines
start at the origin. And then, for some smaller values
of our sphere radius, the purple line is above the red and the blue lines. Eventually though, the blue line
crosses the purple line and the red one does as well. And then, as our radius continues
to increase, the red line crosses the blue line and begins to grow fastest of all
three.
At this point, we can recall the
mathematical equations for the volume of a sphere and its surface area. A sphere of radius 𝑟 has a volume
four-thirds times 𝜋 times 𝑟 cubed, which means that its volume is proportional to
its radius cubed. And then, the surface area of a
sphere is equal to four times 𝜋 times its radius squared, meaning that this
quantity is proportional to 𝑟 squared. Once we know the formulas for 𝑉
and 𝐴, we can take their ratio and find that it’s equal to 𝑟 divided by three,
which tells us that this fraction is proportional to 𝑟. We’ll want to focus on these
proportionalities as we figure out which line on our graph corresponds to which
quantity.
The line that corresponds to 𝑉,
the answer to the first part of our question, will be a line that follows a cubic
progression. Likewise, the line that represents
𝐴 will follow a quadratic progression. And the one that represents 𝑉 over
𝐴 will be linear. Knowing this, we can look at our
graph and note that the red line, even though it starts out below the blue and the
purple line, eventually exceeds them both as the sphere radius increases. So then, by the time it leaves this
graph we see, the red line has the steepest slope or gradiant. In other words, it’s increasing at
the fastest rate. Of the three functional forms we’re
looking for, cubic, quadratic, and linear, we know that it’s the cubic form that
increases in this manner. So, we’ll say that the line on our
graph representing 𝑉 is the red line because as the sphere radius gets larger and
larger, this line increases at the greatest rate.
Next, we want to identify which
line represents the surface area of the sphere. And as we said, this line will be
represented by a quadratic function. Of the lines remaining on our
graph, we see that one, the blue line, has a curve to it as a quadratic function
would, while the purple line looks to be straight. So then, we’ll say that it’s the
blue line that represents the sphere’s surface area as its radius changes. And this leaves us with just one
last line, which we expect will represent 𝑉 divided by 𝐴. As we saw, we expect this line to
be linear, that is, to have a constant slope or gradient. And indeed, we see that the purple
line fits this description. And so, summing up, the red line
represents the volume 𝑉, the blue line represents the surface area 𝐴, and it’s the
purple line that represents 𝑉 divided by 𝐴.
