Two similar cones have volumes 81pi/2 in^3 and 12pi in^3 and slant height of the bigger cone is 7.5 in. Find the integer solution to the height of the similar cone.

To find the integer solution to the height of the similar cone, we need to use the volume and slant height relationships between similar cones.

Let's denote the height of the bigger cone as h_1 and the height of the smaller cone as h_2.

The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14159), r is the radius of the base, and h is the height.

The slant height of a cone (l) can be related to the height (h) and radius (r) by the Pythagorean theorem. We have l^2 = r^2 + h^2.

Given that the volume of the bigger cone is 81π/2 in^3 and the volume of the smaller cone is 12π in^3, we can set up the ratio of volumes:

(Volume of bigger cone) / (Volume of smaller cone) = (81π/2) / (12π)

Simplifying, we get:

(81π/2) / (12π) = 81/2 / 12 = (81/2) * (1/12) = 81/24 = 27/8

Since the ratio of the volumes is equal to the cube of the ratio of the heights, we now have:

(h_1 / h_2)^3 = 27/8

To find the integer solution for the height of the similar cone, we need to find a cube that gives us a result close to 27/8. By trial and error, we can find that 3^3 = 27 and 2^3 = 8.

Therefore, the cube root of 27/8 is 3/2.

So, the height ratio is h_1 / h_2 = 3/2.

Given that the slant height of the bigger cone is 7.5 inches, we can set up a proportion using the height and slant height ratios:

(7.5 / h_1) = (7.5 / h_2) * (h_1 / h_2) = (7.5 / h_2) * (3/2)

Since we are looking for the integer solution for the height of the similar cone, we can assume that the height of the smaller cone (h_2) is an integer multiple of 2.

Let's try different values for h_2:

If h_2 = 2, then (7.5 / h_2) * (3/2) = (7.5 / 2) * (3/2) = 11.25
If h_2 = 4, then (7.5 / h_2) * (3/2) = (7.5 / 4) * (3/2) = 5.625
If h_2 = 6, then (7.5 / h_2) * (3/2) = (7.5 / 6) * (3/2) = 3.75

The only integer solution is h_2 = 6.

Therefore, the height of the similar cone (h_2) is 6 inches.