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A cone is generated by rotating triangle $ABC$ around side $\overline{AB}$.  Its total surface area is $\pi$ times what number?

 

 Aug 23, 2023
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If you could envision rotating \(\triangle \text{ABC}\) around \(\overline{\text{AB}}\), then the diagram would look like the following diagram. I realize the diagram is not drawn to scale. As of now, I have not yet worked out how to draw 3-dimensional figures efficiently.

 

 

The surface area of right cones are \(\text{SA}_\text{right cone} = \pi r^2 + \pi rL\) where r is the radius of the right cone and the L is the slant height of the right cone. We already know the radius of this cone, but we can find L with the use of Pythagorean's Theorem.

 

\(L^2 = 1^2 + 3^2 \\ L^2 = 1 + 9 \\ L^2 = 10 \\ L = \sqrt{10} \text{ or } L = -\sqrt{10}\)

 

Since we are dealing with lengths, reject \(L = -\sqrt{10}\). Now that we have found all the necessary lengths, we can find the surface area of this right cone.

 

\(\begin{align*} \text{SA}_\text{right cone} &= \pi r^2 + \pi rL \\ &= \pi * 3^2 + \pi * 3 * \sqrt{10} \\ &= 9\pi + 3\sqrt{10}\pi \\ &= (9 + 3\sqrt{10}) \pi \end{align*} \)

 

The question asks what the surface area is pi times what number? Well, that number would be \(9 + 3\sqrt{10}\).

 Aug 24, 2023

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