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High School Math : High School Math

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Example Questions

Example Question #3 : Other Polyhedrons

Find the volume of the following polyhedron.

Dome

Possible Answers:

$18 \pi \sqrt{3}m^3 + 81 \pi m^3$

$24 \pi \sqrt{3}m^3 + 81 \pi m^3$

$27 \pi \sqrt{3}m^3 + 81 \pi m^3$

$6 \pi \sqrt{3}m^3 + 81 \pi m^3$

$9 \pi \sqrt{3}m^3 + 81 \pi m^3$

Correct answer:

$9 \pi \sqrt{3}m^3 + 81 \pi m^3$

Explanation:

The formula for the volume of the polyhedron is:

V = (cone)+(cylinder)

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

where is the radius of the cone, is the height of the cone, is the radius of the cylinder, and is the height of the cylinder.

Use the formula for a 30-60-90 triangle to find the length of the radius and height of the cone:

$a-a\sqrt{3}-2a$

$3m-3\sqrt{3}m-6m$

Plugging in our values, we get:

$V = \frac{\pi (3m)^2 (3\sqrt{3}m)}{3}+\pi (3m)^2 (9m)$

$V = 9 \pi \sqrt{3}m^3 + 81 \pi m^3$

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Example Question #1 : Other Polyhedrons

Find the volume of the following polyhedron.

Ice_cream_cone

Possible Answers:

$148 \sqrt{2} \pi m^3$

$128 \sqrt{2} \pi m^3$

$108 \sqrt{2} \pi m^3$

$118 \sqrt{2} \pi m^3$

$138 \sqrt{2} \pi m^3$

Correct answer:

$128 \sqrt{2} \pi m^3$

Explanation:

The formula for the volume of the polyhedron is:

$V = (cone) + \frac{1}{2}(sphere)$

$V = \frac{1}{3}(\pi r^2)(h)+\frac{1}{2}\left(\frac{4\pi r^3}{3}\right)$

where is the radius of the polyhedron and is the height of the cone.

Use the formula for a 45-45-90 triangle to find the length of the radius and height:

$a-a-a\sqrt{2}$

$4\sqrt{2}m-4\sqrt{2}m-8m$

Plugging in our values, we get:

$V = \frac{1}{3}(\pi (4\sqrt{2}m)^2)(4\sqrt{2}m)+\frac{1}{2}\left(\frac{4\pi (4\sqrt{2}m)^3}{3}\right)$

$V = 128 \sqrt{2} \pi m^3$

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Example Question #11 : Solid Geometry

Our backyard pool holds 10,000 gallons. Its average depth is 4 feet deep and it is 10 feet long. If there are 7.48 gallons in a cubic foot, how wide is the pool?

Possible Answers:

100 ft

7.48 ft

133 ft

33 ft

30 ft

Correct answer:

33 ft

Explanation:

There are 7.48 gallons in cubic foot. Set up a ratio:

1 ft³ / 7.48 gallons = x cubic feet / 10,000 gallons

Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft³/ 7.48 gallons) = 1336.9 ft³

Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet

Solve for WIDTH:

4 ft x 10 ft x WIDTH = 1336.9 cubic feet

WIDTH = 1336.9 / (4 x 10) = 33.4 ft

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Example Question #12 : Solid Geometry

A cube has a volume of 64cm³. What is the area of one side of the cube?

Possible Answers:

16cm²

16cm³

4cm²

4cm

16cm

Correct answer:

16cm²

Explanation:

The cube has a volume of 64cm³, making the length of one edge 4cm (4 * 4 * 4 = 64).

So the area of one side is 4 * 4 = 16cm²

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Example Question #3 : How To Find The Length Of An Edge Of A Cube

Given that the suface area of a cube is 72, find the length of one of its sides.

Possible Answers:

$3\sqrt{2}$

$\sqrt{6}$

$2\sqrt{3}$

Correct answer:

$2\sqrt{3}$

Explanation:

The standard equation for surface area is

SA=6a^2

where denotes side length. Rearrange the equation in terms of to find the length of a side with the given surface area:

$a=\sqrt{\frac{SA}{6}}=\sqrt{\frac{72}{6}}=\sqrt{12}=2\sqrt{3}$

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Example Question #1 : Cubes

Find the length of an edge of the following cube:

Length_of_edge

The volume of the cube is $27\ m^3$ .

Possible Answers:

$3\ m$

$9\ m$

$12\ m$

$6\ m$

$15\ m$

Correct answer:

$3\ m$

Explanation:

The formula for the volume of a cube is

V=(e)^3 ,

where is the length of the edge of a cube.

Plugging in our values, we get:

$27\ m^3 = (e)^3$

$e = 3\ m$

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Example Question #3 : How To Find The Length Of An Edge Of A Cube

Find the length of an edge of the following cube:

Length_of_edge

The volume of the cube is $64\ m^3$ .

Possible Answers:

$3\ m$

$8\ m$

$4\ m$

$6\ m$

$5\ m$

Correct answer:

$4\ m$

Explanation:

The formula for the volume of a cube is

V=(e)^3 ,

where is the length of the edge of a cube.

Plugging in our values, we get:

$64\ m^3=(e)^3$

$e=4\ m$

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Example Question #1 : How To Find The Length Of An Edge Of A Cube

What is the length of an edge of a cube that has a surface area of 54?

Possible Answers:

Correct answer:

Explanation:

The surface area of a cube can be determined using the following equation:

$SA=6\times l^2$

54=6l^2

$\frac{54}{6}=\frac{6l^2}{6}$

9=l^2

$\sqrt9=\sqrt{l^2}$

3=l

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Example Question #1 : How To Find The Diagonal Of A Cube

Find the length of the diagonal connecting opposite corners of a cube with sides of length $2\sqrt{2}$ .

Possible Answers:

units

$2\sqrt6$ units

units

$6\sqrt{2}$ units

Correct answer:

$2\sqrt6$ units

Explanation:

Find the diagonal of one face of the cube using the Pythagorean Theorem applied to a triangle formed by two sides of that face ( and ) and the diagonal itself ():

a^2 + b^2 = c^2

$(2\sqrt{2})^2 + (2\sqrt{2})^2 = c^2$

$\rightarrow 8 + 8 = c^2$

$\rightarrow 16 = c^2$

$\rightarrow c = 4$

This diagonal is now the base of a new right triangle (call this ). The height of that triangle is an edge of the cube that runs perpendicular to this diagonal (call this ). The third side of the triangle formed by and is a line from one corner of the cube to the other, i.e., the cube's diagonal (call this ). Use the Pythagorean Theorem again with the triangle formed by , , and to find the length of this diagonal.

a^2 + b^2 = c^2

$(2\sqrt{2})^2 + (4)^2 = c^2$

$\rightarrow 8 + 16 = c^2$

$\rightarrow 24 = c^2$

$\rightarrow \sqrt{24} = c$

$\rightarrow c= 2\sqrt{6}$

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Example Question #571 : Geometry

Find the length of the diagonal of the following cube:

Length_of_diagonal

Possible Answers:

$2\ m$

$4\sqrt{2}\ m$

$4\ m$

$8\ m$

$4\sqrt{3}\ m$

Correct answer:

$4\sqrt{2}\ m$

Explanation:

To find the length of the diagonal, use the formula for a 45-45-90 triangle:

$a-a-a\sqrt{2}$

$4\ m-4\ m-4\sqrt{2}\ m$

The length of the diagonal is $4\sqrt{2}\ m$ .

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High School Math : High School Math

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #3 : Other Polyhedrons

Example Question #1 : Other Polyhedrons

Example Question #11 : Solid Geometry

Example Question #12 : Solid Geometry

Example Question #3 : How To Find The Length Of An Edge Of A Cube

Example Question #1 : Cubes

Example Question #3 : How To Find The Length Of An Edge Of A Cube

Example Question #1 : How To Find The Length Of An Edge Of A Cube

Example Question #1 : How To Find The Diagonal Of A Cube

Example Question #571 : Geometry

All High School Math Resources

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