The volume of a cone is 54 pi cubic inches. The radius and height of the cone are both reduced by a factor of 3 to create a new cone. What is the volume of the new cone?
Is the volume of the new cone 18pi cu in
A. 2 pi cu inches
B. 6 pi cu inches
C. 9 pi cu inches
D. 18 pi cu inches
The volumes of similar cones (i.e. all linear dimensions are proportional) are proportional to the cube of the linear ratios.
For example, if a cone has twice the linear dimensions of a similar cone, the volume is 2³=8 times the similar cone.
To find the volume of the new cone, we need to calculate the radius and height of the new cone using the given information and then substitute those values into the volume formula for a cone.
Given:
Volume of the original cone = 54π cubic inches
Let's denote the radius of the original cone as r and the height as h.
Using the volume formula for a cone, we have:
Volume of the original cone = (1/3)πr^2h
So, we can rewrite the equation as:
54π = (1/3)πr^2h
Dividing both sides of the equation by π, we get:
54 = (1/3)r^2h
Since the radius and height of the new cone are reduced by a factor of 3, we can express the radius and height of the new cone as r/3 and h/3, respectively.
Now, we can substitute these values into the formula:
54 = (1/3)(r/3)^2(h/3)
Simplifying further, we have:
54 = (1/27)rh^2
Next, we need to find the value of rh^2. To do this, we can divide both sides of the equation by (1/27):
54/(1/27) = rh^2
54 * 27 = rh^2
1458 = rh^2
Now, let's find the volume of the new cone by substituting the values of r/3 and h/3 into the volume formula:
Volume of the new cone = (1/3)π(r/3)^2(h/3)
Volume of the new cone = (1/27)πr^2h
Volume of the new cone = (1/27)π * r * h^2
Since rh^2 = 1458, we can substitute this value into the equation:
Volume of the new cone = (1/27)π * 1458
Simplifying further, we have:
Volume of the new cone = 54π
So, the volume of the new cone is 54π cubic inches.
Therefore, the correct answer is D. 18π cubic inches.