How to find the length of an edge of a prism - PSAT Math
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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?
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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
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?
Tap to see back →
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
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?
Tap to see back →
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
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.
Tap to see back →
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.
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?
Tap to see back →
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
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?
Tap to see back →
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
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?
Tap to see back →
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
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.
Tap to see back →
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.
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?
Tap to see back →
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
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?
Tap to see back →
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
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.
Tap to see back →
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.
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?
Tap to see back →
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
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?
Tap to see back →
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
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?
Tap to see back →
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
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?
Tap to see back →
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
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.
Tap to see back →
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.
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?
Tap to see back →
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
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.
Tap to see back →
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.
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?
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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
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?
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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