Mensuration

Volume of cylinder = pi*p^2*h. Volume of sphere = (4/3)*pi*q^3. Since h=q, pi*p^2*q = (4/3)*pi*q^3. Dividing by pi*q (assuming q is not 0), p^2 = (4/3)*q^2, which is 3p^2 = 4q^2.

Volume of sphere = (4/3) * pi * r^3. Surface area = 4 * pi * r^2. Setting them equal: (4/3) * pi * r^3 = 4 * pi * r^2. Dividing by 4 * pi * r^2 gives r/3 = 1, so r = 3.

Volume V = pi * r^2 * h. New radius r' = 0.5r, new height h' = 1.5h. New volume V' = pi * (0.5r)^2 * (1.5h) = pi * 0.25r^2 * 1.5h = 0.375 * V. The change is (0.375 - 1) * 100 = -62.5%.

Volume = pi * r^2 * h. 3850 = (22/7) * r^2 * 25. r^2 = (3850 * 7) / (22 * 25) = 49. So r = 7.

Curved surface area = 2 * pi * r * h = 132. Diameter = 7, so radius r = 3.5. 2 * (22/7) * 3.5 * h = 132. 22 * h = 132, so h = 6.

Area = pi*r^2 = 38.5. r^2 = 38.5 * 7 / 22 = 12.25. r = 3.5. Diameter = 2*r = 7.

Area of a semi-circle = (1/2) * pi * r^2 = 308. (1/2) * (22/7) * r^2 = 308. (11/7) * r^2 = 308. r^2 = 308 * 7 / 11 = 28 * 7 = 196. r = 14.

If radius r increases by 31%, the new radius is 1.31r. The new area is pi*(1.31r)^2 = 1.7161 * pi*r^2. The increase is 71.61%.

Volume of hemisphere = 2/3 * pi * r^3 = 2/3 * (22/7) * 7^3 = 2/3 * 22 * 49 = 2156 / 3 = 718.666... cubic cm.

The area of the circle is pi*r^2 = 154. Using pi = 22/7, r^2 = 154 * 7 / 22 = 49, so r = 7. The diameter is 2r = 14. Since the diameter equals the side of the square, the side length is 14. The area of the square is 14^2 = 196.

The volume of a cylinder is V = pi * r^2 * h. Given V = 385 and h = 10, we have 385 = (22/7) * r^2 * 10. Solving for r^2 gives r^2 = (385 * 7) / 220 = 12.25. Taking the square root, r = 3.5 cm.

Radius r = 5 cm, height h = 12 cm. Volume of cone = (1/3) * pi * r^2 * h = (1/3) * pi * 25 * 12 = 100 * pi. Volume of hemisphere = (2/3) * pi * r^3 = (2/3) * pi * 125 = 83.33 * pi. Total volume = 183.33 * pi = 575.96. 576.19 is the closest approximation.

Volume of sphere = 4/3 * pi * 9^3 = 4/3 * pi * 729 = 972 * pi. Volume of wire = pi * r^2 * h = pi * (0.1 cm)^2 * h = pi * 0.01 * h. 972 * pi = 0.01 * pi * h. h = 97200 cm = 972 meters.