Quantity A: SA of a cylinder = 2_πr_2 + 2_πrh_ = 2_π *_ 16 + 2_π_ * 4 * 2 = 48_π_

Quantity B: SA of a rectangular solid = 2_ab_ + 2_bc_ + 2_ac_ = 2 * 3 * 2 + 2 * 2 * 4 + 2 * 3 * 4 = 52

48_π_ is much larger than 52, because π is approximately 3.14.

The volume of any solid figure is . In this case, the volume of the cylinder is and its height is , which means that the area of its base must be . Working backwards, you can figure out that the radius of a circle of area is . The circumference of a circle with a radius of is , which is greater than .

We need the formula for the surface area of a cylinder: SA = 2_πr_2 + 2_πrh_. This formula has π in it, but the answer choices don't. This means we must approximate π. None of the answers are too close to each other so we could really even use 3 here, but it is safest to use 3.14 as an approximate value of π.

Then SA = 2 * 3.14 * 172 + 2 * 3.14 * 17 * 3 ≈ 2137

There is no need to do the actual computations here to find the two volumes. The volume of a cone is exactly 1/3 the volume of a cylinder with the same height and radius. That means the two quantities are equal. The formulas show this relationship as well: volume of a cone = πr_2_h/3 and volume of a cylinder = πr_2_h.

The formula for the surface area of a cylinder is ,

where is the radius and is the height.

The base of the original cylinder would have been πr2, and the outer face would have been 2πrh, where h is the height of the cylinder.

Let's represent the original area with A, the original radius with r, and the new radius with R: therefore, we know πR2 = 4A, or πR2 = 4πr2. Solving for R, we get R = 2r; therefore, the new outer face of the cylinder will have an area of 2πRh or 2π2rh or 4πrh, which is double the original face area; thus the percentage of increase is 100%. (Don't be tricked into thinking it is 200%. That is not the percentage of increase.)

This problem is simple if we remember the surface area formula!

surface area of a cylinder

= 2_πr_2 + 2_πrh_

= 2_π_ * 62 + 2_π_ * 6 *9

= 180_π_