A cone has a radius of 15 cm and a volume of 540 cm3. What is the volume of a similar cone with a radius of 10 cm?
A. 54 cm3
B. 240 cm3
C. 160 cm3
D. 360 cm3
We know that the ratio of the sides is $10/15=2/3$, so the ratio of the volumes of the two cones is $(2/3)^3=8/27$. If the volume of the large cone is 540, then the volume of the small cone is $(8/27)(540)=\boxed{160}$.
To find the volume of a similar cone with a different radius, we can use the formula for the volume of a cone, which is V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height of the cone.
We are given the radius and volume of the first cone, and we need to find the volume of the second cone.
First, let's find the height of the first cone. The volume of the cone is given as 540 cm^3, and the radius is given as 15 cm. We can rearrange the formula to solve for the height:
V = (1/3)πr^2h
540 = (1/3)π(15^2)h
540 = (1/3)(225π)h
540 = (75π)h
h = 540 / (75π)
h ≈ 2.288 cm
Now we can find the volume of the second cone. We are given the radius of the second cone as 10 cm. Using the formula, we have:
V = (1/3)πr^2h
V = (1/3)π(10^2)(2.288)
V = (1/3)π(100)(2.288)
V = (1/3)(100π)(2.288)
V ≈ 240 cm^3
Therefore, the volume of a similar cone with a radius of 10 cm is approximately 240 cm^3, so the correct answer is B. 240 cm^3.
To find the volume of a similar cone with a radius of 10 cm, we can use the property of similar objects. The ratio of the volumes of two similar objects is equal to the cube of the ratio of their corresponding linear dimensions.
The ratio of the radius of the first cone to the radius of the second cone is 15 cm / 10 cm = 1.5.
Since the radius is a linear dimension, the ratio of their volumes will be (1.5)^3 = 3.375.
To find the volume of the second cone, we can multiply the volume of the first cone by this ratio:
Volume of the second cone = 540 cm^3 * 3.375 = 1822.5 cm^3.
Therefore, the volume of a similar cone with a radius of 10 cm is approximately 1822.5 cm^3.
None of the options given match this answer exactly.