Find the total surface and volume of the body obtained by rotating a rectangular trapezoid with a base of 10 cm

Find the total surface and volume of the body obtained by rotating a rectangular trapezoid with a base of 10 cm and 16 cm around its large base, if the height of the trapezoid is exactly 8 cm.

When you rotate a rectangular trapezoid around the smaller side, you get a body of revolution called a truncated cone.

To calculate the volume of a given body of revolution, we use the formula:

V = 1/3 * π * h (R ^ 2 + Rr + r ^ 2);

V = 1/3 * π * 8 (16 ^ 2 + 16 * 10 + 10 ^ 2) = 1/3 * π * 8 (256 + 16 * 10 + 100) = 1/3 * π * 8 * 516 = 1376π = 4320.64 cm3.

The total surface area of ​​a truncated cone is calculated using the following formula:

S = π * (LR + Lr + R ^ 2 + r ^ 2).

To do this, you need to find a generator. Since the trapezoid AVSD is rectangular, the BH is equal to CD and is 8 cm, and the segment AH is equal to the difference in the length of the two bases:

AH = AD – BC;

AH = 16 – 10 = 6 cm.

AB ^ 2 = BH ^ 2 + AH ^ 2;

AB ^ 2 = 8 ^ 2 + 6 ^ 2 = 64 + 36 = 100;

AB = √100 = 10 cm.

S = π (10 * 16 + 10 * 10 + 16 ^ 2 + 10 ^ 2) = π (160 + 100 + 256 + 100) = 616π = 1934.24 cm2.

Answer: the volume of the body is 4320.64 cm3, the total surface area is 1934.24 cm2.