Mensuration

Curved surface area of a cone is pi*r*l. Since areas are equal, pi*r1*l1 = pi*r2*l2. Thus r1/r2 = l2/l1. Given l1/l2 = 3/5, then r1/r2 = 5/3.

Curved Surface Area = 2 * pi * r * h. Ratio = (r1*h1) / (r2*h2) = (4/5) * (2/3) = 8/15.

Volume = pi * r^2 * h. 3080 = 3.142 * (10.5)^2 * h. 3080 = 3.142 * 110.25 * h. 3080 = 346.4 * h. h = 3080 / 346.4 = 8.89 cm.

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

Volume V = (4/3) * pi * r^3 = 10. r^3 = 30 / (4 * pi) = 7.5 / pi approx 2.387. r = cuberoot(2.387) approx 1.336 m.

Volume of hemisphere = (2/3) * pi * r^3 = 20. r^3 = (20 * 3) / (2 * 3.142) = 30 / 3.142 = 9.548. r = cube root of 9.548 = 2.121.

Volume of a sphere = (4/3) * pi * r^3. With r = 2, V = (4/3) * 3.142 * 8 = (32/3) * 3.142 = 10.666 * 3.142 = 33.51466.

The volume of a cylinder is given by V = pi * r^2 * h. Substituting the given values: 330 = 3.142 * r^2 * 4. Solving for r^2 gives 330 / 12.568 = 26.257, and taking the square root gives approximately 5.12 cm.

The volume of a hemisphere is (2/3) * pi * r^3. With r = 6, volume = (2/3) * 3.142 * 216 = 452.448 ml. Subtracting the existing 325 ml gives 127.448 ml, which rounds to 127.45 ml.

Volume = (4/3) * pi * r^3. r = 5 * sqrt(3). r^3 = 125 * 3 * sqrt(3) = 375 * sqrt(3). Volume = (4/3) * pi * 375 * sqrt(3) = 500 * pi * sqrt(3) = 500 * 3.14159 * 1.732 = 2720.69.

Volume of a cone V = (1/3) * pi * r^2 * h. If r becomes 2r and h remains constant, the new volume V' = (1/3) * pi * (2r)^2 * h = 4 * (1/3) * pi * r^2 * h = 4V. The ratio is 4:1.

Volume of cone = (1/3) * pi * r^2 * h_cone. Volume of cylinder = pi * r^2 * h_cyl. Since volumes are equal, (1/3) * h_cone = h_cyl. Given h_cyl = 5, h_cone = 3 * 5 = 15.

Surface area of cube = 6a^2 = 486. a^2 = 81, so a = 9. Volume = a^3 = 9^3 = 729.

Base area = pi * r^2 = 154. r^2 = 154 / (22/7) = 154 * 7 / 22 = 7 * 7 = 49. r = 7.