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.

bigV/smallV = 81/2 / 12 = 81/24 = (3/2)^3

volumes are in the ratio x^3:1 if the sides are in ratio x:1

so,

bigH/smallH = 3/2

so, 7.5/smallH = 1.5, and smallH = 5

To find the height of the similar cone, we can use the ratio of the volumes of the two cones.

The ratio of the volumes of similar cones is given by the cube of the ratio of their corresponding dimensions (height or radius). In this case, since the slant height of the bigger cone is given, we will use the slant height as the corresponding dimension.

Let's denote the height of the similar cone as h.

The ratio of the volumes of the two cones can be expressed as:

(Volume of bigger cone) / (Volume of smaller cone) = (Height of bigger cone / Height of smaller cone)^3

(81π/2) / (12π) = (7.5 / h)^3

Simplifying this equation:

(81/2)/(12) = (7.5/h)^3

(27/4) = (7.5/h)^3

Taking the cube root of both sides:

∛(27/4) = 7.5/h

∛27/∛4 = 7.5/h

3∛3/2 = 7.5/h

Cross multiplying:

3h = 15∛3

h ≈ 15∛3 / 3

Now, let's approximate the value of h:

h ≈ 5∛3

Calculating the value of ∛3 ≈ 1.44 (using a calculator):

h ≈ 5 * 1.44

h ≈ 7.2

Therefore, the integer solution to the height of the similar cone is 7.

To find the integer solution to the height of the similar cone, we need to use the concept of similar figures. Two figures are similar if their corresponding angles are equal and their corresponding sides are proportional.

Let's start by finding the ratio of the volumes of the two cones. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. Since we want the ratio of volumes, we can ignore the (1/3) and π part, giving us:

V1/V2 = (r1^2h1)/(r2^2h2)

Given that V1 = (81π/2) in^3 and V2 = 12π in^3, we can plug in these values:

(81π/2)/(12π) = (r1^2h1)/(r2^2h2)

Simplifying, we get:

(81/2)/(12) = (r1^2h1)/(r2^2h2)

27/8 = (r1^2h1)/(r2^2h2)

Since the cones are similar, the ratio of their slant heights should be the same as the ratio of their heights:

slant height1/slant height2 = height1/height2

Given that the slant height of the bigger cone is 7.5 inches, we have:

7.5/slant height2 = height1/height2

Simplifying, we get:

7.5/slant height2 = height1/slant height2

height1 = 7.5

Now, let's substitute this value of height1 in the equation we derived earlier:

27/8 = (r1^2 * 7.5)/(r2^2 * slant height2)

Simplifying further, we get:

27/8 = 7.5(r1^2)/(slant height2 * r2^2)

We are given that the slant height of the bigger cone is 7.5 inches, so we can substitute this value:

27/8 = 7.5(r1^2)/(7.5 * r2^2)

Simplifying, we get:

27/8 = (r1^2)/(r2^2)

To find the integer solution to the height of the similar cone, we need to determine the ratio of the radii. Since r1/r2 = √(27/8), we can simplify it further:

r1/r2 = √(27/8)
= (√27)/(√8)
= (√9*√3)/(√4*√2)
= 3/2√2

Since the ratio of the radii is 3/2√2, and the slant height of the big cone is 7.5 inches, we can use the Pythagorean theorem to find the height of the big cone:

(7.5)^2 = (2√2)^2 + (h2)^2
56.25 = 4*2 + (h2)^2
(h2)^2 = 56.25 - 8
(h2)^2 = 48.25
h2 = √48.25
h2 ≈ 6.94

Therefore, the integer solution to the height of the similar cone is 6.