DSQ: Calculating the surface area of a cylinder - GMAT Quantitative

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Question

Of a given cylinder and a given cube, which, if either, has the greater surface area?

Statement 1: Both the height of the cylinder and the diameter of its bases are equal to the length of one edge of the cube.

Statement 2: Each face of the cube has as its area four times the square of the radius of the bases of the cylinder.

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Answer

The surface area of a cylinder, given height and radius of the bases , is given by the formula

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

The surface area of a cube, given the length of each edge, is given by the formula

$A ' = 6s^{2}$ .

Assume Statement 1 alone. Then and, since the diameter of a base is , the radius is half this, or $r = \frac{1}{2}s$ . The surface area of the cylinder is, in terms of , equal to

$A =2 \pi s \left ( s+ \frac{1}{2}s \right )= 2 \pi s \cdot \frac{3}{2}s = 3 \pi s^{2}$ .

Since $3 \pi > 6$ , the cylinder has the greater area regardless of the actual measurements.

Assume Statement 2 alone. The cube has six faces with area $4r^{2}$ , so its surface area is six times this, or $6 \cdot 4r^{2} = 24 r^{2}$ . The surface area of the cylinder is $A =2 \pi r (r+h) = 2 \pi r^{2}+ 2 \pi rh$ ; however, without knowing anything aobout the height of the cylinder, we cannot compare the two surface areas.

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Question

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 $\frac{14}{\pi}$ feet.

II) The height is $12\pi$ times longer the radius.

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Answer

Review our statements:

I) The radius of the silo is $\frac{14}{\pi}$ feet.

II) The height is $12\pi$ 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:

$\small SA=2\pi r^2+2\pi r h$

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

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Question

A tin can has a volume of $\small 375\pi in^3$ .

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)

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Answer

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.

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Question

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 $150\pi$ square meters of metal to build the tank.

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Answer

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.

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Question

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.

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Answer

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:

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

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Question

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 $90 ^{\circ }$ .

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Answer

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.

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Question

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.

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Answer

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.

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Question

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.

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Answer

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:

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

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

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Question

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.

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Answer

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:

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

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

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

$A = 558 \pi$

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Question

Give the surface area of a cylinder.

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

Statement 2: Each base has radius 7.

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Answer

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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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.

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Answer

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.

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Question

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.

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Answer

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

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Question

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.

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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:

$A = 2 \pi R (R+H)$ ;

similarly, the surface area of Cylinder 2 is

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

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

$H = \frac{1}{2} h$ , 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:

$A = 2 \pi \cdot 2r (2r+H)$

$A = 4 \pi r (2r+H)$

$A = 4 \pi r \cdot 2r+ 4 \pi r \cdot H$

$A = 8 \pi r^{2}+4 \pi rH$ .

Cylinder 2 has surface area

$a = 2 \pi r(r+2H)$

$a = 2 \pi r \cdot r+2 \pi r \cdot 2H$

$a = 2 \pi r^{2} +4 \pi rH$

$A - a= (8 \pi r^{2}+4 \pi rH) - \left (2 \pi r^{2} +4 \pi rH \right )= 6 \pi r^{2}$ , so , and Cylinder 1 has the greater surface area.

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Question

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.

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Answer

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

$A = 2 \pi RH$

Assume Statement 1 alone. This means

;

multiplying both sides of the inequality by $2 \pi$ , we get

$2 \pi rh <2 \pi RH$ ,

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

$rh =\frac{1}{2} RH$

It follows that , and, again, $2 \pi rh <2 \pi RH$ , or . Cylinder 2 has the greater lateral area.

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Question

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.

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Answer

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

$a = 2 \pi r(r+h)$ ,

and the surface area of Cylinder 2 is

$A = 2 \pi R(R+H)$ .

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

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

$= 2 \pi \cdot 4 \cdot 8$

$= 64 \pi$

The surface area of Cylinder 2 is

$A = 2 \pi R(R+H)$ ,

$= 2 \pi \cdot 7 \cdot 8$

$= 112 \pi$

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.

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Question

Of two given solids - a cylinder and a regular triangular prism - which has the greater surface area?

Statement 1: Each side of a triangular base of the prism has length one third the circumference of a base of the cylinder.

Statement 2: The cylinder and the prism have the same height.

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Answer

Statement 1 alone provides insufficient information. The surface area of a cylinder with bases of radius and height is

$a = 2 \pi r ^{2}+ 2 \pi r h$ .

The surface area of a regular triangular prism with height and whose (equilateral triangular) bases have common sidelength - and perimeter - is

$A = 2 \cdot \frac{S^{2}\sqrt{3}}{4} + 3S \cdot H$ , or

$A = \frac{S^{2}\sqrt{3}}{2} + 3S H$ .

Statement 1 alone tells us that $S = \frac{1}{3} \cdot 2 \pi r$ , or $S = \frac{2 \pi }{3} r$ . However, without any information about the heights, we cannot compare and . Similarly, Statement 2 alone tells us that , but nothing about or .

However, if we combine what we know from Statement 2 with the information from Statement 1, we can answer the question. The surface area of a cylinder or a prism is equal to its lateral area plus the areas of its two congruent bases.

The lateral area of a cylinder or a prism is the height multiplied by the perimeter or circumference of a base. From Statement 1, the circumference of the bases of the cylinder is equal to the perimeter of the bases of the prism, and from Statement 2, the heights are equal. It follows that the lateral areas are equal, and that the figure with the bases that are greater in area has the greater surface area.

For simplicity's sake, we will assume that the circumference of the base of the cylinder is $6 \pi$ , for reasons that will be apparent later; this reasoning works for any circumference. The radius of this base is $6 \pi \div 2 \pi = 3$ , and the area is $\pi \cdot 3^{2} = 9 \pi$ . The perimeter of the triangular base of the prism is also $6 \pi$ , so its sidelength is $2 \pi$ , making its area $\frac{(2 \pi)^{2}\sqrt{3}}{2} = 2 \pi^{2}\sqrt{3}$ , which is greater than $9 \pi$ . This makes the prism the greater in surface area as well.

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Question

Of a given cylinder and a given sphere, which, if either, has the greater surface area?

Statement 1: The height of the cylinder is equal to the radius of the sphere.

Statement 2: The radius of a base of the cylinder is greater than the radius of the sphere.

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Answer

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

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

Let the radius of the sphere be . Then its surface area can be calculated to be

$A = 4 \pi R^{2}$

From Statement 1 alone, , so the surface area of the cylinder can be expressed as

$a = 2 \pi r R + 2 \pi r^{2}$ .

We cannot compare this to the surface area of the sphere without knowing anything about the radius of a base.

Likewise, from Statement 2 alone, we know that , but without knowing anything about the height, we cannot compare the surface areas.

Now assume both statements. Again, from Statement 1,

$a = 2 \pi r R + 2 \pi r^{2}$ .

Since, from Statement 2, , if follows that

$2 \pi R \cdot r >2 \pi R \cdot R$ , or, $2 \pi r R > 2 \pi R ^{2}$ .

It also follows that

$r^{2} > R^{2}$

and

$2 \pi r^{2} > 2 \pi R^{2}$

Therefore, we can add both sides of the inequalities:

$2 \pi r R + 2 \pi r^{2} > 2 \pi R^{2}+ 2 \pi R ^{2}$

$2 \pi r R + 2 \pi r^{2} > 4 \pi R^{2}$

and

,

so the cylinder has the greater surface area of the two solids.

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Question

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.

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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:

$A = 2 \pi R (R+H)$ ;

similarly, the surface area of Cylinder 2 is

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

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

$H = \frac{1}{2} h$ , 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:

$A = 2 \pi \cdot 2r (2r+H)$

$A = 4 \pi r (2r+H)$

$A = 4 \pi r \cdot 2r+ 4 \pi r \cdot H$

$A = 8 \pi r^{2}+4 \pi rH$ .

Cylinder 2 has surface area

$a = 2 \pi r(r+2H)$

$a = 2 \pi r \cdot r+2 \pi r \cdot 2H$

$a = 2 \pi r^{2} +4 \pi rH$

$A - a= (8 \pi r^{2}+4 \pi rH) - \left (2 \pi r^{2} +4 \pi rH \right )= 6 \pi r^{2}$ , so , and Cylinder 1 has the greater surface area.

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Question

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.

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Answer

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

$A = 2 \pi RH$

Assume Statement 1 alone. This means

;

multiplying both sides of the inequality by $2 \pi$ , we get

$2 \pi rh <2 \pi RH$ ,

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

$rh =\frac{1}{2} RH$

It follows that , and, again, $2 \pi rh <2 \pi RH$ , or . Cylinder 2 has the greater lateral area.

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Question

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 $150\pi$ square meters of metal to build the tank.

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Answer

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.

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