Rectangular Solids & Cylinders

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GMAT Quantitative › Rectangular Solids & Cylinders

Questions 1 - 10

What is the length of the edge of a cube?

Its volume is 1,728 cubic meters.
Its surface area is 864 square meters

EACH statement ALONE is sufficient.

Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.

Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.

Statements 1 and 2 TOGETHER are not sufficient.

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

Explanation

Call the sidelength, surface area, and volume of the cube , , and , respectively.

Then

$V = s^{3}$

or, equivalently,

$s = \sqrt[3]{V}$

So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be

$s = \sqrt[3]{1,728} = 12$

Also,

$A = 6s^{2}$

or, equivalently,

$s = \sqrt{\frac{A}{6}}$

Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be

$s = \sqrt{\frac{864}{6}} = \sqrt{144} = 12$

Therefore, the answer is that either statement alone is sufficient.

Ron is making a box in the shape of a cube. He needs to know how much wood he needs. Find the surface area of the box.

I) The diagonal distance across the box will be equivalent to $4\sqrt{3}ft$ .

II) Half the length of one side is .

Either statement is sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Explanation

To find the surface area of a cube, we need the length of one side.

Statement I gives the diagonal, we can use this to find the length of one side.

Statement II gives us a clue about the length of one side; we can use that to find the full length of one side.

The following formula gives us the surface area of a cube:

Use Statement I to find the length of the side with the following formula, where is the diagonal and is the side length:

$d=\sqrt{3}a$

$4\sqrt{3}=\sqrt{3}a$

So, using Statement I, we find the surface area to be

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

Using Statement, we get that the length of one side is two times two:

$2\cdot 2=4ft$

Again, use the surface area formula to get the following:

$SA=6\cdot 4^2=6\cdot 16=84 ft^2$

What is the length of edge of cube ?

(1) $HB=6\sqrt{3}$ .

(2) $\measuredangle {EHB}=45^{\circ}$ .

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Each statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

In order to find the length of an edge, we would need any information about one of the faces of the cube or about the diagonal of the cube.

Statement 1 gives us the length of the diagonal of the cube, since the formula for the diagonal is $d=s\sqrt{3}$ where is the length of an edge of the cube and is the length of the diagonal we are able to find the length of the edge. Therefore statement 1 alone is sufficient.

Statement 2 alones is insufficient, it gives us something we can already tell knowing that ABCDEFGH is a cube.

Statement 1 alone is sufficient.

is a cube. What is the length of edge ?

(1) The volume of the cube is .

(2) The area of face is .

Each statement alone is sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

Like we have previously seen, to find the length of an edge, we need to have information about the other faces or anything else within the cube.

Statement 1 tells us that the volume of the cube is , from this we can find the length of the side of the cube. Statement 1 alone is sufficient.

Statement 2, tells us that the area of ABCD is , similarily, by taking the square root of this number, we can find the length of the edge of the cube.

Therefore each statement alone is sufficient.

Find the volume of the cube.

1. The cube has a diagonal of 17.32 inches.

2. The cube has a surface area of 600 square inches.

Each statement alone is sufficient.

statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 together are not sufficient.

Explanation

To find the volume of a cube, we only need the length of one side. Using statement 1, we can figure out the length of a side based on the diagonal. We can use the figure below to find the ratio of the diagonal to one side. If the we let the length of one side be x, we can use Pythagorean's theorem to find the length of the diagonal. So, in triangle BCD, we have a right triangle, with two sides of length x. We can set up the equation that the length of BD is $\sqrt{x^2+x^2}= \sqrt{2x^2}=x\sqrt2$ .

Then we can see triangle ADB is also a right triangle. Using Pythagorean's theorem we get the length of AB is $\sqrt{x^2 + (x\sqrt2)^2} = \sqrt{x^2+2x^2}= \sqrt{3x^2} = x\sqrt3$ .

Cube

So, if we divide the number from statement 1 by the square root of 3, we get the length of each side of the cube. Doing this, we get $\frac{17.32}{\sqrt3}= 10$ . Thus, we can solve this problem with just the information from statement 1.

Now, we can also check statement 2. If we know the surface area of the cube, we can use that information to find the length of each side of the cube. We know that the surface area of a cube is the sum of the six faces of the cube, which all have equal area and are all squares. We can divide the total surface area by 6 to find the surface area of each square face. So, 600/6 = 100. We know that the area of a square is just the length of one side squared, so we can take the square root of 100 to find that the length of each side is 10. Thus statement 2 is also sufficient to solve this problem.

Therefore, the answer is that either statement alone is sufficient to answer the question.

Of Cylinder 1 and Cylinder 2, which, if either, has the greater surface area?

Statement 1: The radius of the bases of Cylinder 1 is equal to the height of Cylinder 2.

Statement 2: The radius of the bases of Cylinder 2 is equal to the height of Cylinder 1.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$ .

We show that both statements together provide insufficient information by first noting that if the two cylinders have the same height, and their bases have the same radius, their surface areas will be the same.

Now we explore the case in which Cylinder 1 has height 6 and bases with radius 8, and Cylinder 2 has height 8 and bases of radius 6.

The surface area of Cylinder 1 is

$A = 2 \pi \cdot 8 \cdot 6 + 2 \pi \cdot 8^{2}$

$= 2 \pi \cdot 8 \cdot 6 + 2 \pi \cdot 8 \cdot 8$

$=96 \pi + 128 \pi$

$=224 \pi$

The surface area of Cylinder 2 is

$A = 2 \pi \cdot 6 \cdot 8 + 2 \pi \cdot 6^{2}$

$= 2 \pi \cdot 6 \cdot 8 + 2 \pi \cdot 6 \cdot 6$

$=96 \pi + 72\pi$

$=168\pi$

In this scenario, Cylinder 1 has the greater surface area.

Give the surface area of a cylinder.

Statement 1: The circumference of each base is $18 \pi$ .

Statement 2: The height is four greater than the diameter of each base.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi r(r+h)$

Statement 1 gives the circumference of the bases, which can be divided by $2 \pi$ to yield the radius; however, it yields no information about the height, so the surface area cannot be calculated.

Statement 2 gives the relationship between radius and height, but without actual lengths, we cannot give the surface area for certain.

Assume both statements are true. Since, from Statement 1, the circumference of a base is $18 \pi$ , its radius is $18 \pi \div 2 \pi = 9$ ; its diameter is twice this, or 18, and its height is four more than the diameter, or 22. We now know radius and height, and we can use the surface area formula to answer the question:

$A = 2 \pi r(r+h)$

$A = 2 \pi \cdot 9 \cdot (9+22)$

$A = 2 \pi \cdot 9 \cdot 31$

$A = 558 \pi$

Give the surface area of a cylinder.

Statement 1: If the height is added to the radius of a base, the sum is twenty.

Statement 2: If the height is added to the diameter of a base, the sum is thirty.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi r (r+h)$

We can rewrite the statements as a system of equations, keeping in mind that the diameter is twice the radius:

Statement 1:

Statement 2:

Neither statement alone gives the actual radius or height. However, if we subtract both sides of the first equation from the last:

$\left (h + 2r \right ) -\left (h + r \right ) = 30 - 20$

We substitute back in the first equation:

The height and the radius are both known, and the surface area can now be calculated:

$A = 2 \pi r (r+h)$

$A = 2 \pi \cdot 10 \cdot (10+10)= 2 \pi \cdot 10 \cdot 20 = 400 \pi$

Inscribed_cylinder

In the above figure, a cylinder is inscribed inside a cube. What is the surface area of the cylinder?

Statement 1: The volume of the cube is 729.

Statement 2: The surface area of the cube is 486.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$ .

It can be seen from the diagram that if we let be the length of one edge of the cube, then and $r = \frac{1}{2}s$ . The surface area formula can be rewritten as

$A = 2 \pi \cdot \frac{1}{2}s \cdot s + 2 \pi \left ( \frac{1}{2}s \right ) ^{2}$

$A = \pi s ^{2}+ 2 \pi \cdot \frac{1}{4}s ^{2}$

$A = \pi s ^{2}+ \frac{1}{2} \pi s ^{2}$

$A = \frac{3}{2} \pi s ^{2}$

Subsequently, the length of one edge of the cube is sufficient to calculate the surface area of the cylinder.

From Statement 1 alone, the length of an edge of the cube can be calculated using the volume formula:

$s^{3} = V = 729$

$s = \sqrt[3]{729}= 9$

From Statement 2 alone, the length of an edge of the cube can be calculated using the surface area formula:

$6s^{2} = A = 486$

$s^{2}= 486 \div 6 = 81$

$s = \sqrt{81}= 9$

Since can be calculated from either statement alone, so can the surface area of the cylinder:

$A = \frac{3}{2} \pi s ^{2} = \frac{3}{2} \pi \cdot 9 ^{2} = \frac{243}{2} \pi$

Give the surface area of a cylinder.

Statement 1: The circumference of each base is $14 \pi$ .

Statement 2: Each base has radius 7.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation

The surface area of the cylinder can be calculated from radius and height using the formula:

$A = 2 \pi rh + 2 \pi r^{2}$

Statement 1 gives the circumference of the bases, which can be divided by $2 \pi$ to yield the radius; Statement 2 gives the radius outright. However, neither statement yields information about the height, so the surface area cannot be calculated.

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