Difference between revisions of "2017 AMC 12B Problems/Problem 14"
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<math>\textbf{(A)}\ 8\pi \qquad \textbf{(B)}\ \frac{28\pi}{3} \qquad \textbf{(C)}\ 12\pi \qquad \textbf{(D)}\ 14\pi \qquad \textbf{(E)}\ \frac{44\pi}{3}</math> | <math>\textbf{(A)}\ 8\pi \qquad \textbf{(B)}\ \frac{28\pi}{3} \qquad \textbf{(C)}\ 12\pi \qquad \textbf{(D)}\ 14\pi \qquad \textbf{(E)}\ \frac{44\pi}{3}</math> | ||
| − | ==Solution== | + | ==Solution 1== |
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| + | The top cone has radius 2 and height 4 so it has volume <math> \dfrac{1}{3} \pi (2)^2 \times 4 </math>. | ||
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| + | The frustum is made up by taking away a small cone of radius 1, height 4 from a large cone of radius 2, height 8, so it has volume | ||
| + | <math> \dfrac{1}{3} \pi (2)^2 \times 8 - \dfrac{1}{3} \pi (1)^2 \times 4</math>. | ||
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| + | Adding, we get <math>\dfrac{1}{3} \pi (16+32-4) = \boxed{ \textbf{E} \ \dfrac{44\pi}{3}}</math>. | ||
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| + | Solution by: SilverLion | ||
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| + | ==Solution 2== | ||
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| + | Find the volume of the cone with the method in Solution 1. The volume of the frustrum is <math>\pi\int_{0}^{4} \left(-\frac{1}{4}x+2\right)^2 dx=\frac{28\pi}{3}</math> | ||
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| + | Adding, we get <math>\dfrac{44\pi}{3}</math>. | ||
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| + | Solution by IdentityChaos2020 | ||
{{AMC12 box|year=2017|ab=B|num-b=13|num-a=15}} | {{AMC12 box|year=2017|ab=B|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 00:31, 25 January 2023
Problem
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
Solution 1
The top cone has radius 2 and height 4 so it has volume 
.
The frustum is made up by taking away a small cone of radius 1, height 4 from a large cone of radius 2, height 8, so it has volume
.
Adding, we get 
.
Solution by: SilverLion
Solution 2
Find the volume of the cone with the method in Solution 1. The volume of the frustrum is 
Adding, we get 
.
Solution by IdentityChaos2020
| 2017 AMC 12B (Problems • Answer Key • Resources) | |
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Followed by Problem 15 |
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