Solid Geometry - GRE Quantitative Reasoning

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.

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.

Surface area of a rectangular solid

= 2_lw_ + 2_lh_ + 2_wh_

= 2(6)(4) + 2(6)(3) + 2(4)(3)

= 108

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

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 , which in this case is .

To begin, you must solve for the height of the original cylinder. We know:

For our values, we know:

Now, divide both sides by :

So, if we have a new radius of , our volume will be:

The surface area of a rectangular prism with sides , , and is given as:

.

Two sides are known; it does not matter how they are designated, but for this problem let and , with as the unknown side. This yields equality:

Now that the three dimensions are known, it's possible to calculate the volume:

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:

.

Remember, the formula for the volume of a rectangular prism is width times height times length:

Now, let's solve the word problem for each of these values. We know that . If length is double the width, then the length must be 6 units. If the height is two-thirds the length, then the height must be 4:

Multiply all three values together to solve for the volume:

The volume of the rectangular prism is units cubed.

First, we must ascertain the length of each side. Based on our initial data, we know that the 6 faces of the cube will have a surface area of 6x2. This yields the equation:

6x2 = 486, which simplifies to: x2 = 81; x = 9.

Therefore, each side has a length of 9. Imagine the cube is centered on the origin. This means its "front-left-bottom corner" will be at (–4.5, –4.5, 4.5) and its "rear-right-top corner" will be at (4.5, 4.5, –4.5). To find the distance between these, we use the three-dimensional distance formula:

d = √((x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2)

For our data, this will be:

√( (–4.5 – 4.5)2 + (–4.5 – 4.5)2 + (4.5 + 4.5)2) =

√( (–9)2 + (–9)2 + (9)2) = √(81 + 81 + 81) = √(243) =

√(3 * 81) = √(3) * √(81) = 9√(3)

The shortest length between any two non-adjacent corners will be the diagonal of the smallest face of the rectangular box. The smallest face of the rectangular box is a six-inch by six-inch square. The diagonal of a six-inch square is .

The diagonal length of a cube is found by a form of the distance formula that is akin to the Pythagorean Theorem, though with an additional dimension added to it. It is:

, or , or

Now, if the the value of is , we get simply

Since the volume of the smaller cubes with edges, , is 27, we have:

.

The large cube has edges .

So the surface area of the large cube is:

.