Mensuration

Volume of water = 600cm * 400cm * 1cm = 240,000 cm^3. Cylinder volume = pi * r^2 * h. 240,000 = 3.14 * 20^2 * h. h = 240,000 / (3.14 * 400) = 240,000 / 1256 = 191.07 cm (approx 190.9).

Volume of a cylinder = pi * r^2 * h. Using pi = 22/7, V = (22/7) * 14 * 14 * 7 = 22 * 14 * 14 = 4312 cm^3.

The ratio of surface areas of two spheres is r1^2/r2^2 = 1/4, so the ratio of radii r1/r2 = 1/2. The ratio of volumes is (r1/r2)^3 = (1/2)^3 = 1/8.

Surface area of a sphere is 4*pi*r^2. Cutting it into two hemispheres creates two new circular faces, each with area pi*r^2. Total surface area of one piece = (1/2 * 4*pi*r^2) + pi*r^2 = 2*pi*r^2 + pi*r^2 = 3*pi*r^2. With r=6, area = 3 * pi * 36 = 108*pi.

The volume of a rectangular pyramid is (1/3) * base_area * height. Base area = 15 * 4 = 60. Volume = (1/3) * 60 * 3 = 60 m^3.

The volume of a cylinder is V = pi * r^2 * h. If r becomes 5r, the new volume is V' = pi * (5r)^2 * h = 25 * pi * r^2 * h = 25V.

Curved surface area of a cylinder = 2 * pi * r * h. 440 = 2 * (22/7) * r * 10. 440 = (440/7) * r. Therefore, r = 7.

Cutting a sphere into two halves creates two new circular faces, each with area pi * r^2. Diameter = 17.5, so r = 8.75. Increase = 2 * pi * (8.75)^2 = 2 * 3.14159 * 76.5625 = 481.25.

Large circle radius R = 14. Area = pi * 14^2 = 196*pi. Smaller circles have diameter 14, so radius r = 7. Area of two smaller circles = 2 * (pi * 7^2) = 98*pi. Remaining area = 196*pi - 98*pi = 98*pi = 98 * 22/7 = 308.

Volume of cone = (1/3)pi*r^2*h = (1/3)*pi(3.5)^2*12 = 49*pi. Volume of cylinder = pi*R^2*H = pi*7^2*H = 49*pi*H. Setting 49*pi = 49*pi*H gives H = 1 cm.

Volume of sphere = 4/3 * pi * r^3 = 4/3 * pi * 27 = 36 * pi. Rise in water level h in a cylinder: pi * R^2 * h = 36 * pi. With R = 10, 100 * h = 36, so h = 0.36 cm.

Volume of cylinder = pi * r^2 * h = pi * 7^2 * 9 = 441 * pi. Volume of cone = (1/3) * pi * r^2 * h_cone. Since radii are equal, 441 * pi = (1/3) * pi * 49 * h_cone. 441 = (49/3) * h_cone. h_cone = 441 * 3 / 49 = 9 * 3 = 27 cm.

Let the breadth of the rectangle be B, so the length is B + 3. The perimeter is 2(L + B) = 50, which simplifies to 2B + 3 = 25, giving B = 11 cm and L = 14 cm. The area of the rectangle is 14 * 11 = 154 square centimeters. Since the area of the circle is also 154, we have pi * r^2 = 154, which yields r = 7 cm, and thus the diameter is 14 cm.

Circumference = 2 * pi * r = 22. So, r = 22 / (2 * pi) = 7 / 2 = 3.5 cm. Area = pi * r^2 = (22/7) * (3.5)^2 = (22/7) * 12.25 = 38.5 sq. cm.

Area of path = pi * (R^2 - r^2) = (22/7) * (23^2 - 14^2) = (22/7) * (529 - 196) = (22/7) * 333 = 1046.57 sq m. Cost = 1046.57 * 20 = 20931.4. Closest option is 20930.