The diameter of two similar cones are in the ration 5 to 6. If the volume of the smaller cone is 125 PI cubic centimeters and the diameter of the larger cone is 12 centimeters, what is the height of the larger cone?
the diameters are in the ratio 5:6
the larger is 12, so the smaller is 10
the smaller cone has height h=3, so the larger cone has height 3*(6/5) = 18/5
To find the height of the larger cone, we need to first determine the diameter of the smaller cone. Since the diameters of the two cones are in the ratio 5 to 6 and we are given the diameter of the larger cone (12 centimeters), we can set up the following proportion:
(12 cm)/(x cm) = (6)/(5)
Cross-multiplying, we get:
12 * 5 = x * 6
Simplifying this equation, we have:
60 = 6x
Now, divide both sides of the equation by 6 to solve for x:
x = 10
So, the diameter of the smaller cone is 10 centimeters.
Next, we can find the height of the larger cone using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
We know the volume of the smaller cone is 125π cubic centimeters, and its radius can be found by dividing its diameter by 2 (since radius = diameter/2).
V = (1/3) * π * (5^2) * h
125π = (1/3) * π * 25 * h
125 = (1/3) * 25 * h
Multiplying both sides of the equation by 3, we have:
375 = 25h
Dividing both sides of the equation by 25, we find:
h = 15
Therefore, the height of the larger cone is 15 centimeters.