Question

一种冰淇淋新品由一个杯子和冰淇淋组成。杯子形状是一个高 $4$ 英寸的直圆锥台，底部底面直径为 $2$ 英寸，顶部底面直径为 $4$ 英寸，杯内装满实心冰淇淋；此外上面还有一个高 $4$ 英寸的实心冰淇淋圆锥，其底面在下方，正好是圆锥台的上底面。冰淇淋的总体积是多少立方英寸？

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?

$8\pi$

$\dfrac{28\pi}{3}$

$12\pi$

$14\pi$

$\dfrac{44\pi}{3}$

答案：E

知识点：圆锥体积相似

难度评级：1530

解答：

将圆锥台的侧边延长至一点，由相似三角形可知，该圆锥台等于一个半径为 $2$ 、高为 $8$ 的圆锥减去一个半径为 $1$ 、高为 $4$ 的圆锥。因此圆锥台的体积为顶部半径为 $2$ 、高为 $4$ 的圆锥体积为 $\tfrac13 \pi (2^2)(4) = \tfrac{16}{3}\pi$ 。总体积为 $\tfrac{28}{3}\pi + \tfrac{16}{3}\pi = \tfrac{44}{3}\pi$ 。 $\begin{aligned} &\tfrac13 \pi (2^2)(8) \\ &\quad {}- \tfrac13 \pi (1^2)(4) \\ &\quad {}= \tfrac{32}{3}\pi - \tfrac{4}{3}\pi \\ &\quad {}= \tfrac{28}{3}\pi. \end{aligned}$

所以正确答案是 E。

Extending the frustum's sides to a point, similar triangles show the frustum equals a cone of radius $2$ and height $8$ minus a cone of radius $1$ and height $4:$ $\begin{aligned} &\tfrac13 \pi (2^2)(8) \\ &\quad {}- \tfrac13 \pi (1^2)(4) \\ &\quad {}= \tfrac{32}{3}\pi - \tfrac{4}{3}\pi \\ &\quad {}= \tfrac{28}{3}\pi. \end{aligned}$ The top cone of radius $2$ and height $4$ adds $\tfrac13 \pi (2^2)(4) = \tfrac{16}{3}\pi.$ The total is $\tfrac{28}{3}\pi + \tfrac{16}{3}\pi = \tfrac{44}{3}\pi.$

Thus, the correct answer is E.

2017 AMC 12B 第 14 题

答案：E

难度评级：1530

解答：

其他年份的第 14 题