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.)

We need the formula for the surface area of a cylinder: SA = 2πr² + 2πrh. 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

surface area of a cylinder

= 2πr² + 2πrh

= 2π * 6² + 2π * 6 *9

= 180π

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 = πr²h/3 and volume of a cylinder = πr²h. 

The volume of any solid figure is \(\displaystyle base \times height\). In this case, the volume of the cylinder is \(\displaystyle 200\pi\) and its height is \(\displaystyle 8\), which means that the area of its base must be \(\displaystyle 25\pi\). Working backwards, you can figure out that the radius of a circle of area \(\displaystyle 25\pi\) is \(\displaystyle 5\). The circumference of a circle with a radius of \(\displaystyle 5\) is \(\displaystyle 10\pi\), which is greater than \(\displaystyle 10\).

The formula for the surface area of a cylinder is \(\displaystyle \dpi{100} 2\pi r^{2}+ 2\pi rh\),

where \(\displaystyle r\) is the radius and \(\displaystyle h\) is the height.

\(\displaystyle \dpi{100} SA=2\pi (3)^{2}+2\pi (3)(4)\)

\(\displaystyle \dpi{100} =42\pi\)

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

\(\displaystyle SA = 2\pi r^2 +2\pi rh = 2\pi *16+2\pi *4*8 = 32\pi +64\pi = 96\pi\)

Quantity A: SA of a cylinder = 2πr² + 2πrh = 2π * 16 + 2π * 4 * 2 = 48π

Quantity B: SA of a rectangular solid = 2ab + 2bc + 2ac = 2 * 3 * 2 + 2 * 2 * 4 + 2 * 3 * 4 = 52

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

circumference = πd

d = 2r

volume of cylinder = πr²h

r = 8, h = 4

volume = 256π

The maximum cylindrical base can have a diameter of 6 and therefore a radius of 3. The formula for the volume of a cylinder is \(\displaystyle \dpi{100} \small \pi r^{2}h\), which in this case is \(\displaystyle \dpi{100} \small 3\times 3\times 12\times \pi\).