Prisms - ISEE Upper Level Quantitative Reasoning

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

Answer

The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^{2}+9t^{2} +16t^{2})$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( 48t^{2} +18t ^{2}+24 t^{2})$

$S = 2( 90 t^{2})$

$S = 180 t^{2}$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

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Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

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Question

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Answer

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

Now that we have all our measurements, plug them in and solve:

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

Answer

The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^{2}+9t^{2} +16t^{2})$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( 48t^{2} +18t ^{2}+24 t^{2})$

$S = 2( 90 t^{2})$

$S = 180 t^{2}$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

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Question

Find the surface area of a non-cubic prism with the following measurements:

Answer

The surface area of a non-cubic prism can be determined using the equation:

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Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

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Question

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

Answer

The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^{2}+9t^{2} +16t^{2})$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( 48t^{2} +18t ^{2}+24 t^{2})$

$S = 2( 90 t^{2})$

$S = 180 t^{2}$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

Compare your answer with the correct one above

Question

Find the surface area of a non-cubic prism with the following measurements:

Answer

The surface area of a non-cubic prism can be determined using the equation:

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

Compare your answer with the correct one above

Question

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Answer

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

Now that we have all our measurements, plug them in and solve:

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

Compare your answer with the correct one above

Question

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Answer

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

Now that we have all our measurements, plug them in and solve:

Compare your answer with the correct one above

Question

Find the surface area of a non-cubic prism with the following measurements:

Answer

The surface area of a non-cubic prism can be determined using the equation:

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

The volume of the solid is equal to the product of its length, width, and height, as follows:

$V = 15 \cdot 15 \cdot x =225 x$ .

Compare your answer with the correct one above

Question

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Answer

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

Now that we have all our measurements, plug them in and solve:

Compare your answer with the correct one above

Question

Find the surface area of a non-cubic prism with the following measurements:

Answer

The surface area of a non-cubic prism can be determined using the equation:

Compare your answer with the correct one above

Question

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Answer

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

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Question

A large crate in the shape of a rectangular prism has dimensions 5 feet by 4 feet by 12 feet. Give its volume in cubic yards.

Answer

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{12}{3}$

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{4}{1}$

$V = \frac{80}{9 } = 8 \frac{8}{9 } \textrm{ yd}^{3}$

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