A cone has a radius of 30 cm and a volume of 5400 cm3. What is the volume of a similar, smaller cone that has a radius of 10 cm?
We can use the fact that the ratio of volumes for similar cones is the cube of the ratio of their linear dimensions (height, radius, etc.).
Let V1 be the volume of the larger cone and V2 be the volume of the smaller cone. Let r1 be the radius of the larger cone and r2 be the radius of the smaller cone.
We know V1 = 5400 and r1 = 30. We want to find V2 when r2 = 10.
We can set up the ratio of volumes:
V1/V2 = (r1/r2)^3
Substituting in the known values, we have:
5400/V2 = (30/10)^3
Simplifying, we get:
5400/V2 = 27
Multiplying both sides by V2 and dividing by 27, we get:
V2 = 5400/27 = 200
Therefore, the volume of the smaller cone with radius 10 cm is 200 cm^3.
To find the volume of a similar, smaller cone, we can use the concept of similarity. Similar figures have proportional dimensions.
The ratio of the radii of the two cones is 10 cm / 30 cm = 1/3.
Since the volumes of similar figures are proportional to the cube of the corresponding dimensions, we can use the ratio of the radii to find the ratio of the volumes.
(1/3)^3 = 1/27
So, the smaller cone has a volume that is 1/27 times the volume of the larger cone.
Thus, the volume of the smaller cone is:
(1/27) * 5400 cm3 = 200 cm3
Therefore, the volume of the smaller cone is 200 cm3.