GMAT Math : DSQ: Calculating the volume of a tetrahedron

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Example Question #1 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, , and , and its apex at the point ; Pyramid 2 has as its base the square with vertices at the origin, , and  , and its apex at the point  . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1:   and 

Statement 2:   

Possible Answers:

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

BOTH statements TOGETHER are insufficient to answer the question. 

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

The volume of a pyramid is one third the product of its height and the area its base. The two pyramids have the same base, so the pyramid with the greater height will have the greater volume (and if their heights are equal, their volumes are equal).

Pyramid 1 is shown below:

Pyramid

The base of the pyramid is on the -plane, so the height of the pyramid  is the perpendicular distance from apex  to this plane; this is the -coordinate, . The base of the pyramid is a square of sidelength 10, so its area is the square of 10, or 100. This makes the volume of Pyramid 1 

Similarly, the volume of Pyramid 2 is

Therefore, the problem asks us to determine which of  and  is the greater.

Assume Statement 1 alone. Since , we can multiply all expressions by  to get a range for the volume of Pyramid 1:

Similarly, since , we can multiply all expressions by 36 to get a range of values for the volume of Pyramid 2:

Since the two ranges share values, it cannot be determined for certain which pyramid has the greater volume.

Assume Statement 2 alone. Then, since  and  , it easily follows that 

,

and, subsequently, Pyramid 2 has the greater volume.

Example Question #2 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid_2

Note: Figure NOT drawn to scale, but you may assume  and .

In the above figure, a pyramid with a rectangular base is inscribed inside a rectangular prism; its vertices are . What is the volume of the pyramid? 

Statement 1: 30-60-90 triangle  has area .

Statement 2: Isosceles right triangle  has area 50.

Possible Answers:

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.

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 insufficient 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.

Explanation:

The volume of the pyramid is one third the product of height  and the area of its rectangular base, which is ; that is,

Assume Statement 1 alone.   has area , which is half the product of the length of shorter leg  and longer leg . Also, by the 30-60-90 Theorem, , so, combining these statements,

, and 

However, we do not have any way of finding out , so the volume cannot be calculated.

Assume Statement 2 alone.  is isosceles, so ; again, since the area of a right triangle is half the product of the lengths of its legs, 

However, we have no way of finding out .

The two statements together give all three of , and , so the volume can be calculated as 

Example Question #3 : Dsq: Calculating The Volume Of A Tetrahedron

A solid in three-dimensional coordinate space has four vertices, at points , and  for some positive values of  . What is the volume of the solid?

Statement 1: 

Statement 2: 

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

 

The figure described is the triangular pyramid, or tetrahedron, in the coordinate three-space below.

Tetrahedron

The base of the pyramid can be seen as a triangle with the three known coordinates , and , and the area of its base is half the product of the lengths of its legs, which is 

.

The volume of the pyramid is one third the product of the area of its base, which is 48, and its height, which is the perpendicular distance from the unknown point to the base. Since the base is entirely within the -plane, this distance is the -coordinate of the apex, which is . Therefore, the only thing that is needed to determine the volume of the pyramid is ; this information is provided in Statement 2, but not Statement 1.

Example Question #4 : Dsq: Calculating The Volume Of A Tetrahedron

Tetra_1

In the above diagram, a tetrahedron - a triangular pyramid - with vertices  is shown inside a cube. Give the volume of the tetrahedron.

Statement 1: The perimeter of Square  is 16.

Statement 2: The area of  is 8.

Possible Answers:

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.

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. 

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The volume of the pyramid is one third the product of height  and the area of its base, which in turn, since here it is a right triangle, is half the product of the lengths  and  of its legs. Since the prism in the figure is a cube, the three lengths are equal, so we can set each to . The volume of the pyramid is

Therefore, knowing the length of one edge of the cube is sufficient to determine the volume of the pyramid.

Assume Statement 1 alone. Since the perimeter of Square  is 16, each side of the square, and each edge of the cube has one fourth this measure, or 4.

Assume Statement 2 alone.  has congruent legs, each of measure ; since its area is 8,  can be found as follows:

From either statement alone, the length of each side of the cube, and, subsequently, the volume of the pyramid, can be calculated.

Example Question #5 : Dsq: Calculating The Volume Of A Tetrahedron

Tetra_1

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a tetrahedron, or triangular pyramid. What is the volume of the tetrahedron?

Statement 1:  is an isosceles triangle with area 64.

Statement 2:  is an equilateral triangle with perimeter 48.

Possible Answers:

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

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:

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

Explanation:

Each statement gives enough information about one triangle to determine its area, its angles, and its sidelengths, but no information about the other three triangles is given except for one side. 

Assume both statements are known.  is an isosceles triangle with area 64. Since , we can find this common sidelength using the area formula for a triangle, with these lengths as height and base:

.

This is the length of both  and 

By the 45-45-90 Theorem,  has length  times this, or 

Since  is an equilateral triangle, . Since  is a right triangle, , and , the triangle is also isosceles, and ; by a similar argument, 

The volume of the pyramid can be calculated. Its base, which is congruent to , has area 64, and its height is ; multiply one third by their product to get the volume.

Example Question #6 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, , and , and its apex at the point ; Pyramid 2 has as its base the square with vertices at the origin, , and  , and its apex at the point  . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1:  

Statement 2:  

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

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

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.

Correct answer:

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

Explanation:

The volume of a pyramid is one third the product of its height and the area its base. 

Pyramid 1 is shown below:

Pyramid

The base of the pyramid is on the -plane, so the height of the pyramid  is the perpendicular distance from apex  to this plane; this is the -coordinate, . The base of the pyramid is a square of sidelength 10, so its area is the square of 10, or 100. This makes the volume of Pyramid 1 

Similarly, the volume of Pyramid 2 is 

The problem therefore asks us which, if either, of  to  is the greater quantity.

Assume Statement 1 alone. If , then , and

 

Since , it follows that , and  - that is, Pyramid 2 has the greater volume.

Statement 2 alone gives insufficient information. We take two sets of values of  and  that add up to 25:

Case 1: 

In this case, Pyramid 2 has the greater height and the greater base area, so it easily follows that Pyramid 2 has the greater volume.

Case 2: 

Then the volume of Pyramid 1 is 

and that of Pyramid 2 is 

This makes Pyramid 1 the greater in volume.

Example Question #7 : Dsq: Calculating The Volume Of A Tetrahedron

Pyramid_2

Note: Figure NOT drawn to scale, but you may assume .

In the above figure, a pyramid with a rectangular base is inscribed inside a rectangular prism; its vertices are . What is the volume of the pyramid? 

Statement 1: The hypotenuse  of 30-60-90 triangle  has length 16.

Statement 2: The hypotenuse  of 45-45-90 right triangle  has length .

Possible Answers:

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. 

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.

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.

Explanation:

The volume of the pyramid is one third the product of height  and the area of its rectangular base, which is ; that is,

Assume Statement 1 alone.  is a 30-60-90 triangle with a hypotenuse of length 16. By the 30-60-90 Triangle Theorem, short leg  has length half this, or 8, and long leg  has length  times that of , or . However, the length of  cannot be determined.

Assume Statement 2 alone.  is a  45-45-90 right triangle with a hypotenuse of length . By the 45-45-90 Theorem, its legs  and  each have length  divided by , which is ; however, the length of  cannot be determined.

From the two statements together, we can determine that  and , and calculate the volume:

.

Example Question #8 : Dsq: Calculating The Volume Of A Tetrahedron

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices . What is the volume of the tetrahedron?

Statement 1: Isosceles right triangle  has area 32.

Statement 2: 

Possible Answers:

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.

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. 

Correct answer:

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

Explanation:

The volume of the pyramid is one third the product of the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or  and . Therefore, 

or 

From Statement 1 alone, we know  is isosceles and has area 32; therefore, its common leg length can be determined using the area formula:

Therefore, . However, nothing can be determined about .

Statement 2 alone does not give any of the three desired lengths or any information necessary to find them.

However, Statement 2, along with the information from Statement 1, can be used to find . From Statement 2, , and from Statement 1, ; the Pythagorean Theorem can be used to find . Therefore, all three of , and  can be found, and the volume of the pyramid can be calculated.

Example Question #9 : Dsq: Calculating The Volume Of A Tetrahedron

A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.

Give the volume of a regular tetrahedron.

Statement 1: Each edge has length 8.

Statement 2: Each face has area .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question. 

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

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The formula for the volume of a regular tetrahedron given the length of each edge  is 

.

Statement 1 gives  information explicitly. Statement 2 gives the means to find , since, if  is substituted for  in the formula for an equilateral triangle:

,

the value of  can be determined.

Example Question #10 : Dsq: Calculating The Volume Of A Tetrahedron

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices . What is the volume of the tetrahedron?

Statement 1: Rectangle  has area 200.

Statement 2: Rectangle  has area 120.

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The volume of the pyramid is one third the product of the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or  and . Therefore, 

or 

We need to know the values of , and  to find the volume of of the pyramid. We show that the two statements give insufficient information by examining two scenarios.

Case 1: 

Rectangle  has area .

Rectangle  has area .

The volume of the pyramid is 

 

Case 2: 

Rectangle  has area .

Rectangle  has area .

The volume of the pyramid is 

 

In each case, the conditions of both statements are met, but the volumes of the pyramids differ.

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