Solid Geometry - SAT Math
Card 0 of 20
What is the length of the diagonal of a cube with volume of 1728 in3?
What is the length of the diagonal of a cube with volume of 1728 in3?
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 1728, or (taking the cube root of both sides), s = 12.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or, for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (12)2 + (12)2 + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 1728, or (taking the cube root of both sides), s = 12.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or, for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (12)2 + (12)2 + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)
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What is the length of the diagonal of a cube with surface area of 294 in2?
What is the length of the diagonal of a cube with surface area of 294 in2?
The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6_s_2, where s is the length of the cube. For our data, this is:
6_s_2 = 294
_s_2 = 49
(taking the square root of both sides) s = 7
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √((_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √((x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (7)2 + (7)2 + (7)2) = √( 49 + 49 + 49) = √(49 * 3) = 7√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6_s_2, where s is the length of the cube. For our data, this is:
6_s_2 = 294
_s_2 = 49
(taking the square root of both sides) s = 7
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √((_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √((x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (7)2 + (7)2 + (7)2) = √( 49 + 49 + 49) = √(49 * 3) = 7√(3)
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If a cube is 3” on all sides, what is the length of the diagonal of the cube?
If a cube is 3” on all sides, what is the length of the diagonal of the cube?
General formula for the diagonal of a cube if each side of the cube = s
Use Pythagorean Theorem to get the diagonal across the base:
s2 + s2 = h2
And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:
h2 + s2 = d2
s2 + s2 + s2 = d2
3 * s2 = d2
d = √ (3 * s2) = s √3
So, if s = 3 then the answer is 3√3
General formula for the diagonal of a cube if each side of the cube = s
Use Pythagorean Theorem to get the diagonal across the base:
s2 + s2 = h2
And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:
h2 + s2 = d2
s2 + s2 + s2 = d2
3 * s2 = d2
d = √ (3 * s2) = s √3
So, if s = 3 then the answer is 3√3
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A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube?
A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube?
Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.
Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.
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What is the length of the diagonal of a cube with volume of 512 in3?
What is the length of the diagonal of a cube with volume of 512 in3?
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 512, or (taking the cube root of both sides), s = 8.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 512, or (taking the cube root of both sides), s = 8.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)
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A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?
A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?
The volume of a rectangular prism is 
.
We are told that the shortest edge is 3. Let us call this the height.
We now have 
, or 
.
Now we replace variables by known values:
Now we have:
We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:
If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:
We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).
This diagonal is then 
.
The volume of a rectangular prism is
We are told that the shortest edge is 3. Let us call this the height.
We now have
Now we replace variables by known values:
Now we have:
We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:
If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:
We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).
This diagonal is then
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The dimensions of a right, rectangular prism are 4 in x 12 in x 2 ft. What is the diagonal distance of the prism?
The dimensions of a right, rectangular prism are 4 in x 12 in x 2 ft. What is the diagonal distance of the prism?
The problem is simple, but be careful. The units are not equal. First convert the last dimension into inches. There are 12 inches per foot. Therefore, the prism's dimensions really are: 4 in x 12 in x 24 in.
From this point, things are relatively easy. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be:
d = √(42 + 122 + 242) = √(16 + 144 + 576) = √(736) = √(2 * 2 * 2 * 2 * 2 * 23) = 4√(46)
The problem is simple, but be careful. The units are not equal. First convert the last dimension into inches. There are 12 inches per foot. Therefore, the prism's dimensions really are: 4 in x 12 in x 24 in.
From this point, things are relatively easy. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be:
d = √(42 + 122 + 242) = √(16 + 144 + 576) = √(736) = √(2 * 2 * 2 * 2 * 2 * 23) = 4√(46)
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A rectangular prism has length 7, width 4, and height 4. What is the distance from the top back left corner to the bottom front right corner?
A rectangular prism has length 7, width 4, and height 4. What is the distance from the top back left corner to the bottom front right corner?
The diagonal from the top back left corner to the bottom front right corner will be the hypotenuse of a right triangle. The sides of the triangle will be the height of the box and the diagonal through the middle of one of the rectangular faces. We will be able to solve for the length using the Pythagorean Theorem.
To calculate the length of the hypotenuse, we first must find the length of the rectangular diagonal using the sides of the rectangle. This diagonal will be the hypotenuse of a right triangle with sides 7 and 4. Solve for the diagonal length using the Pythagorean Theorem.
Now we can return to our first triangle. We are given the height, 4, and now have the length of the rectangular diagonal. Use these values to solve for the length of the diagonal that connects the top back left corner and the bottom front right corner.
The diagonal from the top back left corner to the bottom front right corner will be the hypotenuse of a right triangle. The sides of the triangle will be the height of the box and the diagonal through the middle of one of the rectangular faces. We will be able to solve for the length using the Pythagorean Theorem.
To calculate the length of the hypotenuse, we first must find the length of the rectangular diagonal using the sides of the rectangle. This diagonal will be the hypotenuse of a right triangle with sides 7 and 4. Solve for the diagonal length using the Pythagorean Theorem.
Now we can return to our first triangle. We are given the height, 4, and now have the length of the rectangular diagonal. Use these values to solve for the length of the diagonal that connects the top back left corner and the bottom front right corner.
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The base of a right, rectangular prism is a square. Its height is three times that of one of the sides of the base. If its overall volume is 375 in3, what is the diagonal distance of the prism?
The base of a right, rectangular prism is a square. Its height is three times that of one of the sides of the base. If its overall volume is 375 in3, what is the diagonal distance of the prism?
First, let's represent our dimensions. We know the bottom could be represented as being x by x. The height is said to be three times one of these dimensions, so let's call it 3_x_. Based on this, we know the dimensions of the prism are x, x, and 3_x_. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 375 in3, we can say:
375 = x * x * 3_x_ or 375 = 3_x_3
Simplifying, we first divide by 3: 125 = _x_3. Taking the cube root of both sides, we find that x = 5.
Now, be careful. The dimensions are not 5, 5, 5. They are (recall) x, x, and 3_x_. If x = 5, this means the dimensions are 5, 5, and 15.
At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be: d = √(52 + 52 + 152) = √(25 + 25 + 225) = √(275) = √(5 * 5 * 11) = 5√(11) in
First, let's represent our dimensions. We know the bottom could be represented as being x by x. The height is said to be three times one of these dimensions, so let's call it 3_x_. Based on this, we know the dimensions of the prism are x, x, and 3_x_. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 375 in3, we can say:
375 = x * x * 3_x_ or 375 = 3_x_3
Simplifying, we first divide by 3: 125 = _x_3. Taking the cube root of both sides, we find that x = 5.
Now, be careful. The dimensions are not 5, 5, 5. They are (recall) x, x, and 3_x_. If x = 5, this means the dimensions are 5, 5, and 15.
At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be: d = √(52 + 52 + 152) = √(25 + 25 + 225) = √(275) = √(5 * 5 * 11) = 5√(11) in
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The base of a right, rectangular prism has one side that is three times the length of the other. Its height is twice the length of the longer side of the base. If its overall volume is 13,122 in3, what is the diagonal distance of the prism?
The base of a right, rectangular prism has one side that is three times the length of the other. Its height is twice the length of the longer side of the base. If its overall volume is 13,122 in3, what is the diagonal distance of the prism?
First, let's represent our dimensions. We know the bottom could be represented as being x by 3_x_. The height is said to be twice the longer dimension, so let's call it 2 * 3_x_, or 6_x_. Based on this, we know the dimensions of the prism are x, 2_x_, and 6_x_. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 13,122 in3, we can say:
13,122 = x * 3_x_ * 6_x_ or 13,122 = 18_x_3
Simplifying, we first divide by 18: 729 = _x_3. Taking the cube root of both sides, we find that x = 9.
Now, be careful. The dimensions are not 9, 9, and 9. They are (recall) x, 3_x_, and 6_x_. If x = 9, this means the dimensions are 9, 27, and 54.
At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be: d = √(92 + 272 + 542) = √(81 + 729 + 2916) = √(3726) = √(2 * 3 * 3 * 3 * 3 * 23) = 9√(46) in.
First, let's represent our dimensions. We know the bottom could be represented as being x by 3_x_. The height is said to be twice the longer dimension, so let's call it 2 * 3_x_, or 6_x_. Based on this, we know the dimensions of the prism are x, 2_x_, and 6_x_. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 13,122 in3, we can say:
13,122 = x * 3_x_ * 6_x_ or 13,122 = 18_x_3
Simplifying, we first divide by 18: 729 = _x_3. Taking the cube root of both sides, we find that x = 9.
Now, be careful. The dimensions are not 9, 9, and 9. They are (recall) x, 3_x_, and 6_x_. If x = 9, this means the dimensions are 9, 27, and 54.
At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, _d_2 = _x_2 + _y_2 + _z_2, or d = √(_x_2 + _y_2 + _z_2)
For our data, this would be: d = √(92 + 272 + 542) = √(81 + 729 + 2916) = √(3726) = √(2 * 3 * 3 * 3 * 3 * 23) = 9√(46) in.
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The above figure depicts a rectangular prism. Give the length of the shortest path from Point A to Point B that lies completely along the surface of the prism.
The above figure depicts a rectangular prism. Give the length of the shortest path from Point A to Point B that lies completely along the surface of the prism.
The shortest path is along two of the surfaces of the prism. There are three possibilities, each of which are shown below with the relevant faces folded out.
The top diagram shows a path along the right and the front. The path is equal in length to that of a diagonal of a rectangle of length and height 12 and 30, so the length of the path can be calculated using the Pythagorean Theorem. Setting 
:
The bottom diagrams show a path along the top and the front and a path along the rear and the bottom. In both cases, the path is equal in length to that of a diagonal of a rectangle of length and height 20 and 22, so the length of the path can be calculated using the Pythagorean Theorem. Setting 
:
The shortest distance between Points A and B along the surface is therefore 
, or, if simplified using the Product of Radicals Rule:
.
The shortest path is along two of the surfaces of the prism. There are three possibilities, each of which are shown below with the relevant faces folded out.
The top diagram shows a path along the right and the front. The path is equal in length to that of a diagonal of a rectangle of length and height 12 and 30, so the length of the path can be calculated using the Pythagorean Theorem. Setting
The bottom diagrams show a path along the top and the front and a path along the rear and the bottom. In both cases, the path is equal in length to that of a diagonal of a rectangle of length and height 20 and 22, so the length of the path can be calculated using the Pythagorean Theorem. Setting
The shortest distance between Points A and B along the surface is therefore
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Find the diameter of a sphere with a surface area of 
.
Find the diameter of a sphere with a surface area of
Write the formula to find the surface area of a sphere.
Substitute the area and solve for the radius.
The diameter is double the radius.
Write the formula to find the surface area of a sphere.
Substitute the area and solve for the radius.
The diameter is double the radius.
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What is the diameter of a sphere if the surface area is 
?
What is the diameter of a sphere if the surface area is
Write the formula for the surface area of a sphere.
Substitute the area and find the radius.
The diameter is double the radius.
Write the formula for the surface area of a sphere.
Substitute the area and find the radius.
The diameter is double the radius.
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What is the diameter of a sphere with a volume of 
?
What is the diameter of a sphere with a volume of
Write the formula for the volume of a sphere.
Substitute the volume.
Multiply by 
on both sides in order to isolate the 
term.
Cube root both sides.
The diameter is double the radius.
The answer is: 
Write the formula for the volume of a sphere.
Substitute the volume.
Multiply by
Cube root both sides.
The diameter is double the radius.
The answer is:
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The number of square units in the surface area of a cube is twice as large as the number of cubic units in its volume. What is the cube's volume, in cubic units?
The number of square units in the surface area of a cube is twice as large as the number of cubic units in its volume. What is the cube's volume, in cubic units?
The number of square units in the surface area of a cube is given by the formula 6s2, where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s3.
Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:
6s2 = 2s3
Subtract 6s2 from both sides.
2s3 – 6s2 = 0
Factor out 2s2 from both terms.
2s2(s – 3) = 0
We must set each factor equal to zero.
2s2 = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.
Set the other factor, s – 3, equal to zero.
s – 3 = 0
Add three to both sides.
s = 3
This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s3, must then be 33, or 27 cubic units.
The answer is 27.
The number of square units in the surface area of a cube is given by the formula 6s2, where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s3.
Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:
6s2 = 2s3
Subtract 6s2 from both sides.
2s3 – 6s2 = 0
Factor out 2s2 from both terms.
2s2(s – 3) = 0
We must set each factor equal to zero.
2s2 = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.
Set the other factor, s – 3, equal to zero.
s – 3 = 0
Add three to both sides.
s = 3
This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s3, must then be 33, or 27 cubic units.
The answer is 27.
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You own a Rubik's cube with a volume of 
. What is the edge length of the cube?
You own a Rubik's cube with a volume of
You own a Rubik's cube with a volume of 
. What is the edge length of the cube?
To solve for edge length, think of the volume of a cube formula:
Now, we have the volume, so just rearrange it to solve for side length:
You own a Rubik's cube with a volume of
To solve for edge length, think of the volume of a cube formula:
Now, we have the volume, so just rearrange it to solve for side length:
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A right rectangular prism has a volume of 64 cubic units. Its dimensions are such that the second dimension is twice the length of the first, and the third is one-fourth the dimension of the second. What are its exact dimensions?
A right rectangular prism has a volume of 64 cubic units. Its dimensions are such that the second dimension is twice the length of the first, and the third is one-fourth the dimension of the second. What are its exact dimensions?
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 2 * Dim1 = 2x
Dim3: (1/4) * Dim2 = (1/4) * 2x = (1/2) * x
More directly stated, therefore, our dimensions are: x, 2x, and 0.5x. Therefore, the volume is x * 2x * 0.5x = 64, which simplifies to x3 = 64. Solving for x, we find x = 4. Therefore, our dimensions are:
x = 4
2x = 8
0.5x = 2
Or: 2 x 4 x 8
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 2 * Dim1 = 2x
Dim3: (1/4) * Dim2 = (1/4) * 2x = (1/2) * x
More directly stated, therefore, our dimensions are: x, 2x, and 0.5x. Therefore, the volume is x * 2x * 0.5x = 64, which simplifies to x3 = 64. Solving for x, we find x = 4. Therefore, our dimensions are:
x = 4
2x = 8
0.5x = 2
Or: 2 x 4 x 8
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A right rectangular prism has a volume of 120 cubic units. Its dimensions are such that the second dimension is three times the length of the first, and the third dimension is five times the dimension of the first. What are its exact dimensions?
A right rectangular prism has a volume of 120 cubic units. Its dimensions are such that the second dimension is three times the length of the first, and the third dimension is five times the dimension of the first. What are its exact dimensions?
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 3 * Dim1 = 3x
Dim3: 5 * Dim1 = 5x
More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x3 = 120 or x3 = 8. Solving for x, we find x = 2. Therefore, our dimensions are:
x = 2
3x = 6
5x = 10
Or: 2 x 6 x 10
Based on our prompt, we can say that the prism has dimensions that can be represented as:
Dim1: x
Dim2: 3 * Dim1 = 3x
Dim3: 5 * Dim1 = 5x
More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x3 = 120 or x3 = 8. Solving for x, we find x = 2. Therefore, our dimensions are:
x = 2
3x = 6
5x = 10
Or: 2 x 6 x 10
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For a box to fit inside the cupboard, the sum of the height and the perimeter of the box must, at most, be 360 cm. If Jenn has a box that has a height of 40 cm and a length of 23 cm, what is the greatest possible width of the box?
For a box to fit inside the cupboard, the sum of the height and the perimeter of the box must, at most, be 360 cm. If Jenn has a box that has a height of 40 cm and a length of 23 cm, what is the greatest possible width of the box?
First we write out the equation we are given. H + (2_L_ +2_W_) = 360. H = 40 and L = 23
40 + (2(23) + 2_W_) = 360
40 + (46 + 2_W_) = 360
46 + 2_W_ = 320
2_W_ = 274
W = 137
First we write out the equation we are given. H + (2_L_ +2_W_) = 360. H = 40 and L = 23
40 + (2(23) + 2_W_) = 360
40 + (46 + 2_W_) = 360
46 + 2_W_ = 320
2_W_ = 274
W = 137
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The volume of a rectangular prism is 80 cm3. The length, width, and height of the prism are each an integer number of cm. If the dimensions form three terms of an arithmetic sequence, find the average of the three dimensions.
The volume of a rectangular prism is 80 cm3. The length, width, and height of the prism are each an integer number of cm. If the dimensions form three terms of an arithmetic sequence, find the average of the three dimensions.
Method 1:
Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8. The average is 5.
Method 2:
Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:
x3 + 3dx2 + 2d2x - 80 = 0
Find an integer value of x that creates an integer solution for d. Try x=1 and we see the equation 1 + 3d + 2d2 - 80 = 0 or 2d2 + 3d -79 = 0. The determinant of this quadratic is 641, which is not a perfect square. Therefore, d is not an integer when x=1.
Try x=2 and we see the equation 8 + 12d + 4d2 - 80 = 0 or d2 + 3d - 18 = 0. This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3. Since a negative value of d will result in negative dimensions of the prism, d must equal 3. Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8. The average is 5.
Method 1:
Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8. The average is 5.
Method 2:
Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:
x3 + 3dx2 + 2d2x - 80 = 0
Find an integer value of x that creates an integer solution for d. Try x=1 and we see the equation 1 + 3d + 2d2 - 80 = 0 or 2d2 + 3d -79 = 0. The determinant of this quadratic is 641, which is not a perfect square. Therefore, d is not an integer when x=1.
Try x=2 and we see the equation 8 + 12d + 4d2 - 80 = 0 or d2 + 3d - 18 = 0. This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3. Since a negative value of d will result in negative dimensions of the prism, d must equal 3. Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8. The average is 5.
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