ShmoopTube

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Here’s your shmoop du jour...

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The volume of Wyatt’s perfectly spherical head is 36 pi inches cubed.

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What is its surface area in centimeters squared?

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For the sake of the problem, we can just imagine that Wyatt is sans ears or nose.

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And here are the potential answers...

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So… the question LOOKS like a beast –…

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but it’s actually just a really simple conversion one – from inches to centimeters.

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First some translation work is needed in calculating the surface area.

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The key ingredient we need here is r, or radius, which we can then plug in to the formula which

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they give us as 4 pi r squared.

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We can plug in 36 pi into the formula to get

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36 pi equals 4 thirds pi r-cubed.

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We simplify and solve for r.

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So… we multiply both sides by 3 fourths to get rid of that 4 thirds thing

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and we have 27 pi equals pi r cubed.

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Then we just divide both sides by pi to get 27 equals r cubed.

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That 27 is kind of a magic number and should always trigger that green light in your brain

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that it’s the cube of 3.

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In this case, it’s no exception… because that’s exactly what we’re after.

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R cubed is 27, so r is 3.

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Now we can plug the 3 into the formula they give us

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for the surface area of a sphere and that gets written as:

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Area is 4 pi r squared

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Or 4 pi 3 squared

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Or 4 pi times 9

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Or 4 times 9 here is 36.

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36 pi....inches squared.

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Now we just convert from inches to centimeters and we’re home free.

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BUT WAIT, there’s more! A dirty little trick at play here.

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Surface area is in inches SQUARED, not just inches.

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They give us that one inch is 2.5 centimeters ...but 1 SQUARE inch doesn't equal 2.5 centimeters...

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…it equals the quantity 2.5 centimeters SQUARED.

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2.5 squared equals 6.25. So 1 square inch equals 6.25 centimeters squared.

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We can then take 36 pi inches squared and multiply it by 6.25, which equals 225 pi square centimeters.

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Looks like our answer is D!

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As in “Dome head.”

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Another way we could've done this is converted the initial radius of 3 inches to centimeters

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first and then plugged in the converted radius into the surface area formula.

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1 inch equals 2.5 centimeters, so the radius is approximately 7.5 centimeters, or 15 over 2.

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Surface area equals 4 pi times 15 over 2 squared.

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15 over 2 squared equals 225 over 4.

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The 4's cancel out on top and bottom, so the area equals 225 pi centimeters squared,

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just like the answer we got above.

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Now that we know the answer, it’s seems as plain as the nose on Wyatt’s face.

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Oh. Sorry, Wyatt.

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