Mensuration

Volume V = (4/3) * pi * r^3. Taking logarithmic differentiation, dV/V = 3 * dr/r. If dV/V = 9%, then 3 * dr/r = 9%, so dr/r = 3%.

The volume of a cone is V = (1/3) * pi * r^2 * h. If the radius increases by 50%, the new radius is 1.5r. The new volume is (1/3) * pi * (1.5r)^2 * h = 2.25 * V. The increase is 2.25 - 1 = 1.25, or 125%.

Using similar triangles, the radius r and height h of the cylinder satisfy r/h = (5-r)/h_cone_remaining. With h=2r, height of cone=12, radius of cone=5, we get r/(5-r) = 12/5, leading to r = 60/17. Volume = pi * r^2 * h = pi * (60/17)^2 * (120/17) approx 127.5.

Cube side = s. Volume = s^3. Max cylinder has height s and radius s/2. Volume = pi * (s/2)^2 * s = pi * s^3 / 4. Volume = 3.14159 * s^3 / 4 = 0.7854 * s^3. Leftover = 1 - 0.7854 = 0.2146, which is 21%.

Volume V = (1/3) * pi * r^2 * h. If r becomes 2r, the new volume V' = (1/3) * pi * (2r)^2 * h' = (1/3) * pi * 4r^2 * h'. Given V' = 3V, then 4r^2 * h' = 3 * r^2 * h, so h/h' = 4/3.

The volume of the original spheres is (4/3)pi(1^3 + x^3 + 8^3). The new sphere has a diameter of 18, so radius R = 9, and volume (4/3)pi(9^3). Equating volumes: 1 + x^3 + 512 = 729, so x^3 = 216, which means x = 6. The surface area of the sphere with radius 6 is 4 * pi * 6^2 = 144pi.

Curved Surface Area (CSA) = pi * r * l. Ratio of CSA1/CSA2 = (r1 * l1) / (r2 * l2) = (r1/r2) * (l1/l2) = (2/3) * (9/4) = 18/12 = 3/2.

Let the side of the square base be 'a'. The surface area is 2a^2 + 4ah = 192. Given h=10, 2a^2 + 40a - 192 = 0, or a^2 + 20a - 96 = 0. Factoring gives (a+24)(a-4) = 0, so a = 4. The volume is a^2 * h = 4^2 * 10 = 160 cc.

The clock loses 16 minutes every 24 hours. From 5 a.m. on day 1 to 10 p.m. on day 3 is 48 hours + 17 hours = 65 hours. The loss is (16/24) * 65 = 43.33 minutes. Since the clock shows 10 p.m., the actual time is 10 p.m. + 43.33 minutes, which is approximately 10:43 p.m. Option B is the closest.

The surface area of a rectangular tank is 2(lw + lh + wh). For dimensions 12, 4, and 3, the area is 2(12*4 + 12*3 + 4*3) = 2(48 + 36 + 12) = 2(96) = 192 square metres. Dividing 192 by 16 gives 12 litres of paint required.

Volume V = Area of base * height. New Volume V' = (2 * Area) * (3 * height) = 6 * (Area * height) = 6V. The ratio of new to old is 6:1.

Volume of cone = (1/3) * pi * r^2 * h = (1/3) * pi * 9 * 10 = 30 * pi approx 94.2. Volume of hemisphere = (2/3) * pi * r^3 = (2/3) * pi * 27 = 18 * pi approx 56.5. Total = 48 * pi approx 150.8.

Cylinder volume = pi * r^2 * h = pi * 3.5^2 * 7 = pi * 12.25 * 7 = 85.75 * pi = 269.39. Let radii be x and 2x. Sum of volumes = 4/3 * pi * x^3 + 4/3 * pi * (2x)^3 = 4/3 * pi * (x^3 + 8x^3) = 4/3 * pi * 9x^3 = 12 * pi * x^3. 12 * pi * x^3 = 85.75 * pi. x^3 = 85.75 / 12 = 7.1458. x = 1.926. Smaller sphere volume = 4/3 * pi * 1.926^3 = 4/3 * pi * 7.1458 = 29.94.

Surface area of a room is proportional to the square of the side length. If side increases by 50% (factor 1.5), area increases by (1.5)^2 = 2.25. New paint = 12 * 2.25 = 27. Extra paint = 27 - 12 = 15.