Question

Math

Posted 6 months ago

```
Julie is measuring two cones.
Given the base radius $r$ and height $h$ of the first cone, Julie uses the formula
$V=\frac{\pi r^{2} h}{3}$
to compute its volume $V$ to be 6 cubic meters.
The second cone has the same height, but has 2 times the radius. What is its volume?
Choose 1 answer:
(A) 12 cubic meters
(B) $4 \pi$ cubic meters
(C) 24 cubic meters
(D) $8 \pi$ cubic meters
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Given the formula for the volume of a cone $V = \frac{\pi r^2 h}{3}$ and $V = 6$, solve for $h$

step 2

Substitute $V = 6$ into the formula to get $6 = \frac{\pi r^2 h}{3}$

step 3

Multiply both sides by $3$ to isolate $\pi r^2 h$ on one side: $18 = \pi r^2 h$

step 4

Divide both sides by $\pi r^2$ to solve for $h$: $h = \frac{18}{\pi r^2}$, with the condition that $r \neq 0$

step 5

For the second cone, the radius is twice that of the first cone, so the new radius is $2r$

step 6

Use the volume formula for the second cone with the new radius: $V = \frac{\pi (2r)^2 h}{3}$

step 7

Substitute $h$ from step 4 into the new volume formula: $V = \frac{\pi (2r)^2 \cdot \frac{18}{\pi r^2}}{3}$

step 8

Simplify the expression by canceling out $\pi$ and $r^2$: $V = \frac{4 \cdot 18}{3}$

step 9

Calculate the new volume: $V = 4 \cdot 6 = 24$ cubic meters

[question 1] Answer

(C) 24 cubic meters

Key Concept

Scaling dimensions in geometric shapes

Explanation

When the radius is doubled, the area of the base is quadrupled because the area is proportional to the square of the radius. Since the height remains the same, the volume is also quadrupled.

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