Question Video: Determining the Ratio between the Volume and Area of a Sphere Physics

The volume 𝑉 of a sphere of radius π‘Ÿ is given by the formula 𝑉 = (4/3)πœ‹π‘ŸΒ³. The surface area 𝐴 of the same sphere is given by 𝐴 = 4πœ‹π‘ŸΒ². Which of the following equations correctly gives the ratio of the volume of the sphere to the surface area of the sphere, 𝑉/𝐴? [A] 𝑉/𝐴 = (1/3)π‘ŸΒ³ [B] 𝑉/𝐴 = (1/3)πœ‹π‘Ÿ [C] 𝑉/𝐴 = (1/4)πœ‹π‘ŸΒ² [D] 𝑉/𝐴 = (1/3)π‘Ÿ [E] 𝑉/𝐴 = (1/3)π‘ŸΒ²

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Video Transcript

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

OKay, so in this exercise, we’re considering a sphere that has a radius π‘Ÿ. We’re 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, we’re 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, let’s see what will cancel out. There’s a factor of four in both numerator and denominator, also a factor of πœ‹. And moreover, we can see that two factors of the radius π‘Ÿ cancel from top and bottom. When we remove all the cancelled factors, we’re left with π‘Ÿ, 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.

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