Question

question 17 of 21
the two cones below are similar. what is the value of x?
image of two cones: left cone (blue) with height 10, radius 3; right cone (green) with height 2, radius x
a. 0.6
b. 0.09
c. 0.3
d. 0.18

Explanation:

Step1: Recall similar figures ratio

For similar cones, the ratio of corresponding linear dimensions (height and radius) is equal. So \(\frac{\text{Height of large cone}}{\text{Height of small cone}}=\frac{\text{Radius of large cone}}{\text{Radius of small cone}}\).
Let height of large cone \(h_1 = 10\), height of small cone \(h_2 = 2\), radius of large cone \(r_1 = 3\), radius of small cone \(r_2 = x\). Then \(\frac{10}{2}=\frac{3}{x}\).

Step2: Solve for x

Cross - multiply: \(10x=2\times3\), so \(10x = 6\). Then \(x=\frac{6}{10}=0.6\)? Wait, no, wait. Wait, maybe I mixed up. Wait, no, the large cone has height 10 and radius 3? Wait, no, maybe the small cone's height is 2, large is 10. So ratio of heights is \(10:2 = 5:1\). So radius of small should be radius of large divided by 5? Wait, no, similar figures: corresponding sides are proportional. So \(\frac{\text{Large height}}{\text{Small height}}=\frac{\text{Large radius}}{\text{Small radius}}\) → \(\frac{10}{2}=\frac{3}{x}\) → \(10x = 6\) → \(x = 0.6\)? But wait, maybe I got the ratio reversed. Wait, maybe small height is 2, large height is 10, so scale factor from small to large is \(10/2 = 5\). So small radius \(x\), large radius 3. So \(x=\frac{3}{5}=0.6\)? Wait, but let's check again.

Wait, similar cones: the ratio of heights is equal to the ratio of radii. So \(\frac{h_1}{h_2}=\frac{r_1}{r_2}\). Here, \(h_1 = 10\), \(h_2 = 2\), \(r_1 = 3\), \(r_2 = x\). So \(\frac{10}{2}=\frac{3}{x}\) → \(10x=6\) → \(x = 0.6\). Wait, but the options have 0.6 as option A. Wait, maybe I made a mistake in the diagram. Wait, maybe the large cone has radius 3 and height 10, small cone has height 2 and radius x. So the proportion is correct. So solving gives x = 0.6.

Answer:

A. 0.6