Solid Geometry - GRE Quantitative Reasoning

Side 1: surface area of the 6 x 7 side is 42

Side 2: surface area of the 7 x 8 side is 56

Side 3: surface area of the 6 x 8 side is 48.

We can add sides 1 and 3 to get 90, so that's not the answer.

We can add sides 1 and 1 to get 84, so that's not the answer.

We can add sides 2 and 3 to get 104, so that's not the answer.

We can add sides 2 and 2 to get 112, so that's not the answer.

This leaves the answer of 92. Any combination of the three sides of the rectangular prism will not give us 92 as the total surface area.

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