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## Example Questions

### Example Question #1 : Dsq: Calculating The Surface Area Of A Cylinder

Jiminy wants to paint one of his silos. One gallon of this paint covers about square feet. How many gallons will he need?

I) The radius of the silo is feet.

II) The height is times longer the radius.

**Possible Answers:**

Either statement alone is sufficient to answer the question.

Neither I nor II is sufficient to answer the question. More information is needed.

Both statements are necessary to answer the question.

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

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

**Correct answer:**

Both statements are necessary to answer the question.

Review our statements:

I) The radius of the silo is feet.

II) The height is times longer the radius

We need to find our surface area in order to find how many gallons we need. Surface area is given by:

So to find the surface area, we need the radius and the height, so both statments are needed here.

### Example Question #2 : Dsq: Calculating The Surface Area Of A Cylinder

A tin can has a volume of .

I) The height of the can is inches.

II) The radius of the base of the can is inches.

What is the surface area of the can? (Assume it is a perfect cylinder)

**Possible Answers:**

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

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

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

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

**Correct answer:**

Either statement is sufficient to answer the question.

To find surface area of a cylinder we need the radius and the height.

If we are given the volume, and either the radius or the height, we can work backwards to find the other dimension.

Since I and II give us the height and the radius, either statement can be used to find the surface area.

### Example Question #3 : Dsq: Calculating The Surface Area Of A Cylinder

The tank of a tanker truck is made by bending sheet metal and then welding on the ends. If the length of the tank is meters, what is its radius?

I) The volume of the tank is .

II) It takes square meters of metal to build the tank.

**Possible Answers:**

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

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

Either statement is sufficient to answer the question.

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.

**Correct answer:**

Either statement is sufficient to answer the question.

To find the radius of a cylinder from either volume or surface area we need the height.

We are given the height in the question.

We are given volume and surface area in the two statements.

Thus, either statement is sufficient.

### Example Question #4 : Dsq: Calculating The Surface Area Of A Cylinder

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

Statement 1: The sum of the height of Cylinder 1 and the radius of one of its bases is equal to the sum of the height of Cylinder 2 and the radius of one of its bases.

Statement 2: The bases of Cylinder 1 and Cylinder 2 have the same cicumference.

**Possible Answers:**

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

EITHER statement ALONE is sufficient to answer the question.

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.

**Correct answer:**

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

We will let and represent the radii of Cylinder 1 and Cylinder 2, respectively, and and represent the heights of Cylinder 1 and Cylinder 2, respectively.

The surface area of Cylinder 1 is

,

and the surface area of Cylinder 2 is

.

Statement 1 alone is insufficient, as can be seen by examining these two cases.

Case 1:

For each cylinder, the sum of the radius and the height is 8 - that is, .

The surface area of Cylinder 1 is

The surface area of Cylinder 2 is

,

Therefore, Cylinder 2 has the greater area.

Case 2:

This simply switches the dimensions of the cylinders, and consequently, it switches the surface areas. Cylinder 1 has the greater surface area.

Each scenario satisfies the condition of Statement 1.

Assume Statement 2 alone. The circumferences of the bases are the same, so, subsequently, the radii are as well. But the heights are also needed, and Statement 2 does not clue us in to the heights.

Assume both statements are true.

By Statement 1, .

By Statement 2, since the circumferences of the bases are equal, so are their radii, so .

By subtraction, it follows that , and . Since the cylinders have the same height and their bases have the same radius, it follows that their surface areas are equal.

### Example Question #5 : Dsq: Calculating The Surface Area Of A Cylinder

Give the surface area of a cylinder.

Statement 1: The circumference of each base is .

Statement 2: Each base has radius 7.

**Possible Answers:**

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.

EITHER statement ALONE is 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.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

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

Statement 1 gives the circumference of the bases, which can be divided by 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.

### Example Question #6 : Dsq: Calculating The Surface Area Of A Cylinder

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

Statement 1: The product of the height of Cylinder 1 and the radius of one of its bases is less than the product of the height of Cylinder 2 and the radius of one of its bases.

Statement 2: The product of the height of Cylinder 2 and the radius of one of its bases is equal to the product of the height of Cylinder 1 and the diameter of one of its bases.

**Possible Answers:**

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.

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.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

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

.

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

Assume Statement 1 alone. This means

;

multiplying both sides of the inequality by , we get

,

or

,

Therefore, Cylinder 2 has the greater lateral area.

Assume Statement 2 alone. Since the diameter of a base of Cylinder 1 is twice its radius, or , this means

or

It follows that , and, again, , or . Cylinder 2 has the greater lateral area.

### Example Question #7 : Dsq: Calculating The Surface Area Of A Cylinder

Give the surface area of a cylinder.

Statement 1: The circumference of each base is .

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

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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

Statement 1 gives the circumference of the bases, which can be divided by 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 , its radius is ; 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:

### Example Question #8 : Dsq: Calculating The Surface Area Of A Cylinder

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

Statement 1: Cylinder 1 has bases with radius twice those of the bases of Cylinder 2.

Statement 2: The height of Cylinder 1 is half that of Cylinder 2.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

We will let and stand for the radii of the bases of Cylinders 1 and 2, respectively, and and stand for their heights.

The surface area of Cylinder1 can be calculated from radius and height using the formula:

;

similarly, the surface area of Cylinder 2 is

Therefore, we are seeking to determine which, if either, is greater, or .

Statement 1 alone tells us that , but without knowing anything about the heights, we cannot compare to . Similarly, Statement 2 tells us that

, or, equivalently, , but without any information about the radii, again, we cannot determine which of and is the greater.

Now assume both statements to be true. Substituting for and for , Cylinder 1 has surface area:

.

Cylinder 2 has surface area

, so , and Cylinder 1 has the greater surface area.

### Example Question #9 : Dsq: Calculating The Surface Area Of A Cylinder

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.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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

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:

We substitute back in the first equation:

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

### Example Question #10 : Dsq: Calculating The Surface Area Of A Cylinder

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.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

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:

.

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

Also, the volume can be calculated using the formula

,

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:

Cylinder 2:

However, their lateral areas differ:

Cylinder 1:

Cylinder 2:

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