Non-Cubic Prisms - High School Math

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Question

The length of a box is 3 times the width. Which of the following gives the length (L inches) in terms of the width (W inches) of the box?

Answer

When reading word problems, there are certain clues that help interpret what is going on. The word “is” generally means “=” and the word “times” means it will be multiplied by something. Therefore, “the length of a box is 3 times the width” gives you the answer: L = 3 x W, or L = 3W.

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Question

The width of a box, in inches, is 5 inches less than three times its length. Which of the following equations gives the width, W inches, in terms of the length, L inches, of the box?

Answer

We notice the width is “5 inches less than three times its width,” so we express W as being three times its width (3L) and 5 inches less than that is 3L minus 5. In this case, W is the dependent and L is the independent variable.

W = 3L - 5

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Question

What is the surface area of a rectangular brick with a length of 12 in, a width of 8 in, and a height of 6 in?

Answer

The formula for the surface area of a rectangular prism is given by:

SA = 2LW + 2WH + 2HL

SA = 2(12 * 8) + 2(8 * 6) + 2(6 * 12)

SA = 2(96) + 2(48) + 2(72)

SA = 192 + 96 + 144

SA = 432 in2

216 in2 is the wrong answer because it is off by a factor of 2

576 in3 is actually the volume, V = L * W * H

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Question

David wants to paint the walls in his bedroom. The floor is covered by a $10\ ft \times 16\ ft$ carpet. The ceiling is $8\ ft$ tall. He selects a paint that will cover $75\ ft^2$ per quart and $300\ ft^2$ per gallon. How much paint should he buy?

Answer

Find the surface area of the walls: SAwalls = 2lh + 2wh, where the height is 8 ft, the width is 10 ft, and the length is 16 ft.

This gives a total surface area of 416 ft2. One gallon covers 300 ft2, and each quart covers 75 ft2, so we need 1 gallon and 2 quarts of paint to cover the walls.

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Question

A box is 5 inches long, 5 inches wide, and 4 inches tall. What is the surface area of the box?

Answer

The box will have six total faces: an identical "top and bottom," and identical "left and right," and an identical "front and back." The total surface area will be the sum of these faces.

Since the six faces consider of three sets of pairs, we can set up the equation as:

$SA=2(\text{top})+2(\text{left})+2(\text{front})$

Each of these faces will correspond to one pair of dimensions. Multiply the pair to get the area of the face.

Substitute the values from the question to solve.

$SA=130\ in^2$

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Question

What is the surface area of an equilateral triangluar prism with edges of 6 in and a height of 12 in?

Let $\sqrt{3}=1.732$ and $\sqrt{2}=1.414$ .

Answer

The surface area of the prism can be broken into three rectangular sides and two equilateral triangular bases.

The area of the sides is given by: $A=lw=12\cdot 6=72\ in^{2}$ , so for all three sides we get $\dpi{100} 216\ in^{2}$ .

The equilateral triangle is also an equiangular triangle by definition, so the base has congruent sides of 6 in and three angles of 60 degrees. We use a special right traingle to figure out the height of the triangle: 30 - 60 - 90. The height is the side opposite the 60 degree angle, so it becomes $3\sqrt{3}$ or 5.196.

The area for a triangle is given by $A=\frac{1}{2}bh$ and since we need two of them we get $A=bh=6\cdot 3\sqrt{3}=31.176\ in^{2}$ .

Therefore the total surface area is $216\ in^{2}+31.176\ in^{2}=247.176\ in^{2}$ .

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Question

Angie is painting a 2 foot cube for a play she is in. She needs $25\hspace{1 mm}mL$ of paint for every square foot she paints. How much paint does she need?

Answer

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

$Area=6(2^2)=6\times 4=24\hspace{1 mm}feet^2$

Now we will determine the amount of paint needed

$24\hspace{1 mm}feet^2\times \frac{25\hspace{1 mm}mL}{1\hspace{1 mm}foot^2}=600\hspace{1 mm}mL$

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Question

Find the surface area of the following triangular prism.

Answer

The formula for the surface area of a triangular prism is:

$SA = 2 (base) + lateral\ area$

$SA = 2 \cdot \left(\frac{1}{2}\right)(l)(w) + (l+w+y)(h)$

Where is the length of the triangle, is the width of the triangle, is the hypotenuse of the triangle, and is the height of the prism

Use the formula for a triangle to solve for the length of the hypotenuse:

$s-s-s\sqrt{2}$

$6-6-6\sqrt{2}$

Plugging in our values, we get:

$SA = (6m)(6m) + (6m+6m+6\sqrt{2}m)(9m)$

$SA = (6m)(6m) + (12m + 6\sqrt{2}m)(9m)$

$SA = 36m^2 + 108m^2 + 54\sqrt{2}m^2 = 144 + 54\sqrt{2} m^2$

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Question

Find the surface area of the following triangular prism.

Answer

The formula for the surface area of a triangular prism is:

$SA = 2\left(\frac{1}{2}\right)(l)(w) + (l+w+y)(h)$

Where is the length of the base, is the width of the base, is the hypotenuse of the base, and is the height of the prism

Use the formula for a triangle to find the length of the hypotenuse:

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

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

Plugging in our values, we get:

$SA = (8m)(8m) + (8m+8m+8\sqrt{2}m)(12m)$

$SA = 256m^2 + 96\sqrt{2}m^2$

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Question

Find the surface area of the following triangular prism.

Answer

The formula for the surface area of an equilateral, triangular prism is:

$SA = 2\left(\frac{s^2\sqrt{3}}{4}\right)+(3s)(h)$

Where is the length of the triangle side and is the length of the height.

Plugging in our values, we get:

$SA = 2\left(\frac{(6m)^2\sqrt{3}}{4}\right)+(3)(6m)(12m)$

$SA = 216 + 18\sqrt{3}m^2$

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Question

A rectangular box has two sides with the following lengths:

and

If it possesses a volume of $84$ cm$^{3}$ , what is the area of its largest side?

Answer

The volume of a rectangular prism is found using the following formula:

$V=l\times w\times h$

If we substitute our known values, then we can solve for the missing side.

$84=3\times 4\times x$

Divide both sides of the equation by 12.

$\frac{84}{12}=\frac{12x}{12}$

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.

$A=l\times w$

$A=4\times 7$

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Question

Find the volume of the following triangular prism.

Answer

The formula for the volume of a triangular prism is:

$V = \frac{1}{2} (l)(w)(h)$

Where is the length of the triangle, is the width of the triangle, and is the height of the prism

Plugging in our values, we get:

$V = \frac{1}{2} (6m)(6m)(9m)$

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Question

Find the volume of the following triangular prism.

Answer

The formula for the volume of a triangular prism is:

$V = \frac{1}{2}(l)(w)(h)$

Where is the length of the base, is the width of the base, and is the height of the prism

Plugging in our values, we get:

$V = \frac{1}{2}(8m)(8m)(12m)$

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Question

Find the volume of the following triangular prism:

Answer

The formula for the volume of an equilateral, triangular prism is:

$V = \left(\frac{s^2\sqrt{3}}{4}\right)(h)$

Where is the length of the triangle side and is the length of the height.

Plugging in our values, we get:

$V = \left(\frac{(6m)^2\sqrt{3}}{4}\right)(12m)$

$V=108\sqrt{3}m^3$

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Question

What is the volume?

Answer

The volume is calculated using the equation:

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