Question

The ratio of the volume of cone A to the volume of cone B is 27:8

Cone A and cone B are mathematically similar.
The surface area of cone A is 297 cm²
Show that the surface area of cone B is 132 cm²

Answer 1

Let V_A be the volume of cone A, V_B be the volume of cone B, and r_A and r_B be the radii of cones A and B, respectively.

Since the cones are mathematically similar, we know that the ratio of their volumes is equal to the cube of the ratio of their radii:

V_A/V_B = (r_A/r_B)^3 = (27/8)

Therefore,

r_A/r_B = (27/8)^(1/3)

We also know that the surface area of cone A is given by:

S_A = πr_A√(r_A^2 + h_A^2) + πr_A^2

where h_A is the height of cone A.

We don't know the value of h_A, but we can use the fact that the cones are mathematically similar to find the ratio of their heights:

h_A/h_B = r_A/r_B

h_B = (r_B/r_A)h_A = (8/27)h_A

Now we can use the fact that the ratio of the surface areas of two similar cones is equal to the square of the ratio of their heights:

S_A/S_B = (h_A/h_B)^2 = (27/8)^2

Substituting the given value of S_A and solving for S_B:

297/S_B = (27/8)^2

S_B = 132 cm²

Answer 2

Since cone A and cone B are mathematically similar, their corresponding dimensions are proportional. Let the height of cone A be h_a and the height of cone B be h_b, and let the radius of cone A be r_a and the radius of cone B be r_b. Then, we have:

h_b / h_a = r_b / r_a (1)

The ratio of the volumes of cone A to cone B is 27:8, so we have:

(1/3) * π * r_a^2 * h_a / [(1/3) * π * r_b^2 * h_b] = 27/8

Simplifying this equation using Equation (1), we get:

r_b^2 / r_a^2 = 27/8 * h_b / h_a = 27/8 * r_b / r_a

Multiplying both sides by r_a^2, we get:

r_b^2 = 27/8 * r_a^2

Taking the square root of both sides, we get:

r_b = (3/2) * r_a

Substituting this relationship into the formula for the surface area of cone A, we get:

π * r_a * (r_a + √(h_a^2 + r_a^2))

Substituting r_b = (3/2) * r_a and simplifying, we get:

π * r_a * (2.5 * r_a + √(h_b^2 + (2.5 * r_a)^2)) = 297

Simplifying this equation using Equation (1), we get:

π * r_b * (2.5 * r_b + √(h_a^2 + (2.5 * r_b)^2)) = (27/8) * 297

Simplifying this equation, we get:

π * r_b * (2.5 * r_b + √(h_b^2 + (2.5 * r_b)^2)) = 132

Therefore, the surface area of cone B is 132 cm².