Cylinders

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GMAT Quantitative › Cylinders

Questions 1 - 10

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:

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

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

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

$A = 558 \pi$

What is the volume of the cylinder?

Statement 1: the cylinder has a radius of 3

Statement 2: the cylinder has a height of 4

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

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

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

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation

The formula for the volume of a cylinder is: $volume = \Pi \ast r^{2}\ast h$

Therefore we need both Statement 1 and 2 to find the volume, so both statements together are sufficient, but neither statement alone is sufficient.

The city of Wilsonville has a small cylindrical water tank in which it keeps an emergency water supply. Give its surface area, to the nearest hundred square feet.

Statement 1: The water tank holds about 37,700 cubic feet of water.

Statement 2: About ten and three fourths gallons of paint, which gets about 350 square feet of coverage per gallon can, will need to be used to paint the tank completely.

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

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

Statement 1 is unhelpful in that it gives the volume, not the surface area, of the tank. The volume of a cylinder depends on two independent values, the height and the area of a base; neither can be determined, so neither can the surface area.

From Statement 2 alone, we can find the surface area. One gallon of paint covers 350 square feet, so, since $10 \frac{3}{4}$ gallons of this paint will cover about

$10 \frac{3}{4} \cdot 350 \approx 3,800$ square feet, the surface area of the tank.

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.

Inscribed_cylinder

In the above figure, a cylinder is inscribed inside a cube. and mark the points of tangency the upper base has with $\overline{BC}$ and $\overline{CD}$ . What is the surface area of the cylinder?

Statement 1: Arc $\widehat{XY}$ has length $5 \pi$ .

Statement 2: Arc $\widehat{XY}$ has degree measure $90 ^{\circ }$ .

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

Assume Statement 1 alone. Since $\widehat{XY}$ has length one fourth the circumference of a base, then each base has circumference $5 \pi \cdot 4 = 20 \pi$ , and radius $r = C \div \left (2\pi \right )= 20 \pi \div \left (2 \pi \right )= 10$ . It follows that each base has area $A = \pi r^{2}= \pi \cdot 10^{2}= 100 \pi$

Also, the diameter is $C \div \pi = 20 \pi \div \pi = 20$ ; it is also the length of each edge, and it is therefore the height. The lateral area is the product of height 20 and circumference $20 \pi$ , or $20 \cdot 20 \pi = 400 \pi$ .

The surface area can now be calculated as the sum of the areas:

$400 \pi + 100 \pi+ 100 \pi= 600 \pi$ .

Statement 2 is actually a redundant statement; since each base is inscribed inside a square, it already follows that $\widehat{XY}$ is one fourth of a circle - that is, a $90^{\circ }$ arc.

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.

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: 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:

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

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

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

Statement 1: The cylinders have the same volume.

Statement 2: The product of the height of Cylinder 1 and the area of its base is equal to the product of the height of Cylinder 2 and the area of its base.

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 volume of a cylinder is the product of its height and the area of its base, so the two statements are actually equivalent. Therefore, we demonstrate that knowing that the volumes are the same is insufficient to determine which cylinder, if either, has the greater lateral area.

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

$a = 2 \pi rh$ .

In this problem we will use and as the dimensions of Cylinder 1 and and as those of Cylinder 2. Therefore, the lateral area of Cylinder 2 will be

$A = 2 \pi RH$

Also, the volume can be calculated using the formula

$v= \pi r^{2} h$ ,

so this will come into play.

Case 1: The cylinders have the same height and their bases have the same radii.

It easily follows that they have the same volume and the same lateral area.

Case 2:

The volumes are the same:

Cylinder 1: $v= \pi r^{2} h = \pi \cdot 4^{2} \cdot 9 = 144 \pi$

Cylinder 2: $V = \pi R^{2}H = \pi \cdot 3^{2} \cdot 16 = 144 \pi$

However, their lateral areas differ:

Cylinder 1: $a = 2 \pi rh = 2 \pi \cdot 4 \cdot 9= 72\pi$

Cylinder 2: $A = 2 \pi RH = 2 \pi \cdot 6 \cdot 16 = 192 \pi$

How much water, in cubic feet, can a cylindrical water tank whose bases have radius 6 feet hold?

Statement 1: The lateral area of the tank is 125.66 square yards.

Statement 2: The tank is 30 feet high.

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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

Explanation

We are given the radius; if we know the height, we can use the formula

$V=\pi r^{2}h$

to calculate the volume of the tank.

The second statement gives us that the tank is 30 feet high. But the first statement gives us the way to find the height by using the lateral area formula.

First we have to convert square yards to square feet by multiplying by 9.

$125.66 \cdot 9 = 1,130.94$

$L.A.= 2\pi rh$

$1,130.94= 2\pi \cdot 6\cdot h= 12\pi h$

$h=\frac{ 1,130.94}{12\pi }=30$

Either way, we now have both radius and height, and we can find the volume:

$V=\pi \cdot 6^{2} \cdot 30 \approx 3392.9$

The answer is that either statement alone is sufficient to answer the question.

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