Solid Geometry - ISEE Upper Level Quantitative Reasoning

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Question

Find the volume of a pyramid with the following measurements:

length: 7in

width: 6in

height: 8in

Answer

To find the volume of a pyramid, we will use the following formula:

$V = $\frac{lwh}{3}$$

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in

width: 6in

height: 8in

So, we get

$V = $\frac{ 7\text{in}$ \cdot 6$\text{in}$ \cdot 8$\text{in}$}{3}$

$V = $$\frac{42\text{in}$^2$ \cdot 8$\text{in}$}{3}$

$V = $$\frac{336\text{in}$^3$}{3}$

$V = $112$\text{in}$^3$$

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Question

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

Answer

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the volume of the box?

Find the volume of a cube via the following:

$V_${cube}=s^3$$=(6in)^3$$=216in^3$$

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Question

A cone has height 240 centimeters; its base has radius 80 centimeters. Give its volume in cubic meters.

Answer

Convert both dimensions from centimeters to meters by dividing by 100:

Height: 240 centimeters = $240 \div 100 = 2.4$ meters.

Radius: 80 centimeters = $80 \div 100 = 0.8$ meters.

Substitute in the volume formula:

$V = $\frac{1}{3}$ \pi $r^{2}$h$

$V = $\frac{1}{3}$ \cdot \pi \cdot 0.8 ^{2} \cdot 2.4$

$V = $\frac{1}{3}$\cdot 0.8 \cdot 0.8 \cdot 2.4 \cdot \pi$

$V =0.512 \pi$

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Question

A cylinder has the following measurements:

Height: 12in
Diameter: 10in

Find the volume.

Answer

To find the volume of a cylinder, we will use the following formula:

$V = \pi \cdot $r^2$ \cdot h$

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 10in. We also know the diameter is two times the radius. Therefore, the radius is 5in.

We know the height of the cylinder is 12in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot $(5$\text{in}$)^2$ \cdot 12$\text{in}$$

$V = \pi \cdot $25$\text{in}$^2$ \cdot 12$\text{in}$$

$V = \pi \cdot $300$\text{in}$^3$$

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Question

Find the volume of a cube with a height of 3in.

Answer

To find the volume of a cube, we will use the following formula:

where a is the length of any side of the cube.

Now, we know the height of the cube is 3in. Because it is a cube, all sides (lengths, widths, height) are the same. That is why we can find any length for the formula.

Knowing this, we can substitute into the formula. We get

$V = $(3$\text{in}$)^3$$

$V = 3$\text{in}$ \cdot 3$\text{in}$ \cdot 3$\text{in}$$

$V = $9$\text{in}$^2$ \cdot 3$\text{in}$$

$V = $27$\text{in}$^3$$

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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

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

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Answer

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet = $4 \times 12 = 48$ inches

Sidelength of the base: 3 feet = $3 \times 12 = 36$ inches

The volume of a pyramid is

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

Since the base is a square, we can replace $B = $s^{2}$$ :

$V = $\frac{1}{3}$ s ^{2}h$

Substitute

$V = $\frac{1}{3}$ \cdot 36 ^{2}\cdot 48$

The pyramid has volume 20,736 cubic inches.

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Question

A foot tall pyramid has a square base measuring feet on each side. What is the volume of the pyramid?

Answer

In order to find the area of a triangle, we use the formula $$\frac{1}{3}$Bh$ . In this case, since the base is a square, we can replace with , so our formula for volume is .

Since the length of each side of the base is feet, we can substitute it in for .

$$\frac{1}{3}$\cdot $(40)^{2}$\cdot h$

We also know that the height is feet, so we can substitute that in for .

$$\frac{1}{3}$\cdot $(40)^{2}$\cdot 300$

This gives us an answer of $160,000 $ft^{3}$$ .

It is important to remember that volume is expressed in units cubed.

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Question

The height of a right pyramid is inches. Its base is a square with sidelength inches. Give its volume in cubic feet.

Answer

Convert each of the measurements from inches to feet by dividing by .

Height: $42 \div 12 = 3 $\frac{1}{2}$$ feet

Sidelength: $24 \div 12 = 2$ feet

The base of the pyramid has area

$B = $s^{2}$ = $2^{2}$ = 4$ square feet.

Substitute $B = 4, h = 3 $\frac{1}{2}$$ into the volume formula:

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

$V = \frac {1}{3}\cdot 4 \cdot 3 $\frac{1}{2}$$

$V = \frac {1}{3} \cdot \frac {4}{1} \cdot $\frac{7}{2}$$

$V = \frac {1}{3} \cdot \frac {2}{1} \cdot $\frac{7}{1}$ = \frac {14}{3} = 4 \frac {2}{3}$ cubic feet

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Question

The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.

Answer

Convert each of the measurements from feet to inches by multiplying by .

Height: $2 \times 12 = 24$ inches

Sidelength of base: $3 \times 12 = 36$ inches

The base of the pyramid has area

$B = $s^{2}$ = $36^{2}$ = 1,296$ square inches.

Substitute into the volume formula:

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

$V = \frac {1}{3} \cdot 1,296 \cdot 24 =10,368$ cubic inches

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Question

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

Answer

The perimeter of the square base, feet, is equivalent to $3 \times 12 = 36$ inches; divide by to get the sidelength of the base - and the height: $36 \div 4 = 9$ inches.

The area of the base is therefore $B = $s^{2}$ = $9^{2}$ = 81$ square inches.

In the formula for the volume of a pyramid, substitute :

$V = $\frac{1}{3}$ Bh = $\frac{1}{3}$\cdot 81 \cdot 9 = 243$ cubic inches.

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Question

What is the volume of a pyramid with the following measurements?

$length=7; width=6;height=9;slant\ length=11$

Answer

The volume of a pyramid can be determined using the following equation:

$V=\frac{1}{3}$lwh=\frac{1}{3}$(7)(6)(9)=126$

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Question

A right regular pyramid with volume has its vertices at the points

where .

Evaluate .

Answer

The pyramid has a square base that is units by units, and its height is units, as can be seen from this diagram,

The square base has area $B = $n^{2}$$ ; the pyramid has volume $V = $\frac{1}{3}$ \cdot $n^{2}$ \cdot n = $\frac{1}{3}$ $n^{3}$$

Since the volume is 1,000, we can set this equal to 1,000 and solve for :

$$\frac{1}{3}$ $n^{3}$ = 1,000$

$n = \sqrt[3]{3,000} = \sqrt[3]{1,000} \cdot \sqrt[3]{3 } = 10 \sqrt[3]{3 }$

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Question

Find the volume of a pyramid with the following measurements:

length = 4in

width = 3in

height = 5in

Answer

To find the volume of a pyramid, we will use the following formula:

$V = $\frac{l \cdot w \cdot h}{3}$$

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the base of the pyramid has a length of 4in. We also know the base of the pyramid has a width of 3in. We also know the pyramid has a height of 5in.

Knowing this, we can substitute into the formula. We get

$V = $\frac{4\text{in}$ \cdot 3$\text{in}$ \cdot 5$\text{in}$}{3}$

$V = $$\frac{60\text{in}$^3$}{3}$

$V = $20$\text{in}$^3$$

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Question

Find the volume of a pyramid with the following measurements:

length = 4cm

width = 9cm

height = 8cm

Answer

To find the volume of a pyramid, we will use the following formula:

$V = $\frac{l \cdot w \cdot h}{3}$$

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the following measurements:

length = 4cm

width = 9cm

height = 8cm

Knowing this, we can substitute into the formula. We get

$V = $\frac{4\text{cm}$ \cdot 9$\text{cm}$ \cdot 8$\text{cm}$}{3}$

$V = $$\frac{288\text{cm}$^3$}{3}$

$V = $96$\text{cm}$^3$$

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Question

Find the volume of a pyramid with the following measurements: length= in, width= in, height= in

Answer

To find the volume of a pyramid, we will use the following formula:

$V = $\frac{lwh}{3}$$

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in

width: 6in

height: 8in

So, we get

$V = $\frac{ 7\text{in}$ \cdot 6$\text{in}$ \cdot 8$\text{in}$}{3}$

$V = $\frac{42\text{in}$ \cdot 8$\text{in}$}{3}$

$V = $\frac{336\text{in}$}{3}$

$V = $112$\text{in}$^3$$

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Question

The volume of a cube is . What is the length of an edge of the cube?

Answer

Let be the length of an edge of the cube. The volume of a cube can be determined by the equation:

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