A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm³. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of metal sheet used in making it. (Use π = 3.14)

Let R and r be the radii of the top and bottom circular ends of the frustum respectively. Let h be the height of the frustum. We are given:

R = 20 cm
r = 12 cm
Volume (V) = 12308.8 cm³
π = 3.14

The volume of a frustum of a cone is given by the formula:

V = (1/3)πh(R² + Rr + r²)

Substituting the given values:

12308.8 = (1/3) * 3.14 * h * (20² + 20*12 + 12²)
12308.8 = (1/3) * 3.14 * h * (400 + 240 + 144)
12308.8 = (1/3) * 3.14 * h * 784
12308.8 = 821.1733h
h = 12308.8 / 821.1733
h ≈ 15 cm

Therefore, the height of the bucket is approximately 15 cm.

Now, let's find the slant height (l) of the frustum using the formula:

l = √(h² + (R - r)²)
l = √(15² + (20 - 12)²)
l = √(225 + 64)
l = √289
l = 17 cm

The area of the metal sheet used in making the bucket is the sum of the areas of the circular base and the curved surface area of the frustum. The curved surface area of a frustum is given by:

A = π(R + r)l
A = 3.14 * (20 + 12) * 17
A = 3.14 * 32 * 17
A = 1700.48 cm²

Area of the bottom circular end = πr² = 3.14 * 12² = 452.16 cm²

Total area of the metal sheet used = Curved surface area + Area of bottom
Total area = 1700.48 + 452.16 = 2152.64 cm²

Therefore, the height of the bucket is approximately 15 cm, and the area of the metal sheet used in making it is approximately 2152.64 cm².