Rectangular Solids & Cylinders - GMAT Quantitative

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Question

What is the length of the edge of a cube?

Its volume is 1,728 cubic meters.

Its surface area is 864 square meters

Answer

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.

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Question

A sphere is inscribed inside a cube. What is the volume of the sphere?

Statement 1: The surface area of the cube is 216.

Statement 2: The volume of the cube is 216.

Answer

The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula $V = \frac{4}{3} \pi r^{3}$ can be used to find the volume of the sphere.

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Question

What is the length of edge of cube ?

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

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

Answer

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.

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Question

is a cube. What is the length of edge ?

(1) The volume of the cube is .

(2) The area of face is .

Answer

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.

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Question

Find the length of an edge of the cube.

The volume of the cube is .

The surface area of the cube is .

Answer

Statement 1: Use the volume formula for a cube to solve for the side length.

where represents the length of the edge

$a=\sqrt[3]{125}=5cm$

Statement 2: Use the surface area formula for a cube to solve for the side length.

Each statement alone is sufficient to answer the question.

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Question

Given a right square pyramid and a right cone, which, if either, has greater volume?

Statement 1: The two have the same height.

Statement 2: The base of the cone and the base of the pyramid have the same area.

Answer

Given base area height , the volume of either a pyramid or a cone is equal to

$V = \frac{1}{3} Bh$ .

If we know the heights of the two figures are the same, we still need to know which has the greater base area in order to know which has the greater volume, so Statement 1 is insufficient. By a similar argument, Statement 2 is insufficient.

If we know both heights are equal and that both bases have equal area, then we know that the volumes are equal.

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Question

While deep-sea diving, Al found a rectangular treasure chest. Before he opens it, he wants to find the volume of the chest. What is it?

I) Al measured the chest's longest side to be twice its smallest side and one-and-a-half times the length of the medium side.

II) Al found that the chest's shortest side is 2.5ft long.

Answer

To find the volume of a rectangular solid, use the following formula:

$V=l\cdot w\cdot h$

Statement I relates the length, width and height of the treasure chest.

Statement II gives us the length of the shortest side of the treasure chest.

Use Statement II and the relationships described in Statement I to find each side, then multiply them all together to get your answer.

If the shortest side is 2.5ft, then the longest side must be 5ft.

If the longest side is 5ft, then the medium side can be found via the following:

$5\cdot \frac{1}{1.5}=3\frac{1}{3}ft$

So the volume is as follows:

$5\cdot 3\frac{1}{3}\cdot 2.5=41\frac{2}{3}ft^3$

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Question

Find the volume of a rectangular prism.

Statement 1: , ,

Statement 2: The diagonal is 6.

Answer

Statement 1: , ,

The height can be easily solved for by substituting the length and width. Height is 2, and the volume can be calculated by the following formula.

Statement 2: The diagonal is 6.

Write the equation of the diagonal in terms of length, width, and height.

$d=\sqrt{L^2+W^2+H^2}$

Although the lengths from Statement 1 will yield a diagonal of six, there are multiple combinations of length, width, and height which will also give a diagonal of six. The product of these three dimensions may or may not give a volume of six. Statement 2 has insufficient information to calculate the volume.

Therefore:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

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Question

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 .

Answer

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:

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

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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.

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.

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.

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

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.

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)

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

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.

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 $90 ^{\circ }$ .

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.

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.

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.

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.

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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