Question Video: Understanding the Graphical Representation of the Volume and Area of a Sphere

The graph shows how the volume, 𝑉, the surface area, 𝐴, and the volume-to-surface-area ratio, 𝑉/𝐴, vary with the radius of a sphere. Which line represents 𝑉? Which line represents 𝐴? Which line represents 𝑉/𝐴?

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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 π‘Ÿ has a volume four-thirds times πœ‹ times π‘Ÿ 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œ‹ times its radius squared, meaning that this quantity is proportional to π‘Ÿ squared. Once we know the formulas for 𝑉 and 𝐴, we can take their ratio and find that it’s equal to π‘Ÿ divided by three, which tells us that this fraction is proportional to π‘Ÿ. 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.

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