Solid Geometry - GRE Quantitative Reasoning

First, find the area of the base of the cyclinder:

and multiply that by two, since there are two sides with this measurement: .

Then, you find the width of the rectangular portion (label portion) of the prism by finding the circumference of the cylinder:

. This is then multiplied by the height of the cylinder to find the area of the rectanuglar portion of the cylinder: .

Finally, add all sides together and round: .

Find the area of the triangular sides first:

Since there are two sides of this area, we multiply the area by 2:

Next find the area of the rectangular regions. Two of them have the width of 3 feet and a length of 7 feet, while the last one has a width measurement of feet and a length of 7 feet. Multiply and add all other sides:

.

Lastly, add the triangular sides to the rectangular sides and round:

.

The box is 2 times as long as it is high, so H = L/2. It is also 4 times as wide as it is long, so W = 4_L_. Now we need volume = L * W * H = L * 4_L_ * L/2 = 2_L_3.

The surface area of a cube is merely the sum of the surface areas of the squares that make up its faces. Therefore, the surface area equation understandably is:

, where is the side length of any one side of the cube. For our values, we know:

Solving for , we get:

or

Now, the volume of a cube is defined by the simple equation:

For , this is:

Recall that the volume of a cube is defined by the equation:

, where is the side length of the cube.

Therefore, if we know that , we can solve:

This means that .

Now, if we triple to , the new volume of our cube will be:

Recall that the equation for the surface area of a cube is merely derived from the fact that the cube's faces are made up of squares. It is therefore:

For our values, this is:

Solving for , we get:

, so

Now, the volume of a cube is merely:

Therefore, for , this value is:

To find the volume of a cube, you multiple your side length 3 times (s*s*s).

To find the side length from the volume, you find the cube root which gives you 4

.

Doubling the side gives you 8

.

The volume of the new cube would then be 512

.

The volume of a sphere is represented by the equation . Set this equation equal to the volume given and solve for r:

Therefore, the radius of the sphere is 3.

The formula for the volume of a sphere is

where is the radius of the sphere.

Therefore,

, giving us .

For a sphere to fit into the rectangular prism, its dimensions are constrained by the prism's smallest side, which forms its diameter. Therefore, the largest sphere will have a diameter of , and a radius of .

The volume of a sphere is given as:

And thus the volume of the largest possible sphere to fit into this prism is

surface area of a cylinder

= 2_πr_2 + 2_πrh_

= 2_π_ * 62 + 2_π_ * 6 *9

= 180_π_

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 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!

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

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 formula for the surface area of a cylinder is ,

where is the radius and is the height.

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 .

circumference = πd

d = 2r

volume of cylinder = πr2h

r = 8, h = 4

volume = 256π