Other Polyhedrons - ACT Math

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Question

If we have a regular (the triangles are equilateral) triangular prism of volume and the side length of the triangle on either face is , what is the length of the prism? Write your answer in terms of a decimal rounded to the nearest hundredth.

Answer

The volume of a triangular prism can be simply stated as V = AL, where A is the area of the triangular face and L is the length. We have the volume already and we need to figure out the area of the triangle from the side length.

The area of a triangle is bh/2, and we are given the base/side length: 3 in. We also have a formula for the height of an equilateral triangle, $b\frac{\sqrt{3}}{2}$ . Calculating the area of the triangle we get 2.5981 in^2, so now we just have to plug our numbers into the volume formula.

$V = A\cdot L\Rightarrow 32 in^3 = 2.5981 in^2 \cdot L$

$L = \frac {32}{2.5981} in = 12.32 in$

And that is the final answer.

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Question

The surface area of a cylinder is given by $2\pi rh+2\pi r^{2}$ , where is the radius and is the height. If a cylinder has a surface area of $20\pi$ $cm^{2}$ and a height of , what is its radius?

Answer

The fastest way to solve this problem is to plug all of the answer choices in for and look for an output of $20\pi$ .

Alternatively, one could set the surface area formula equal to $20\pi$ (knowing that ), but this would require solving a quadratic.

$20\pi =2\pi r(3)+2\pi r^2$

$0=6\pi r +2\pi r^2-20\pi$

Rearranging this we get

$2\pi r^2+6 \pi r-20\pi=0$

Now factoring out a $2\pi$ we get:

$2\pi (r^2+3r-10)=0$

$2\pi(r+5)(r-2)=0$

$r+5=0 \rightarrow r=-5$

$r-2=0 \rightarrow r=2$

Since we cannot have a negative length our answer is .

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Question

If a half cylinder with a height of 5 and semicircular bases with a radius of 2, what is the surface area?

Answer

The surface area of the half cylinder will consist of the lateral area and the base area.

There are three parts to the surface area:

Rectangular region, semi-circular bases of the half cylinder, and outer face of the cylinder.

The cross section of the half cylinder is a rectangle. Find the area of the rectangle. The diameter of the semicircle represents the width of the rectangle, which is double the radius. The length of the rectangle is the height of the cylinder.

$A_{\textup{rectangle}}=LW=(4)(5)=20$

Next, find the area of a semicircular bases.

$A_{\textup{semicircular base}} = \frac{\pi r^2}{2} = \frac{\pi (2^2)}{2}= 2\pi$

Since there are two semicircular bases of the semi-cylinder, the total area of the semicircular bases is $4\pi$ .

$A_{\textup{ TWO semicircular bases}} = 4\pi$

Find the area of the outer region. The area of the outer region is the half circumference multiplied by the height of the cylinder.

$A_{\textup{outer region}}= \frac{2\pi r}{2} \times h = 2\pi (5)= 10\pi$

Sum the areas of the rectangle, the two circular bases, and the outer region to find the lateral area.

$\sum A=20 + 4\pi + 10\pi = 20+14\pi$

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Question

A cube has a surface area of $150\textup{m}^2$ . What is its volume?

Answer

First, find the side lengths of the cube.

Recall that the surface area of the cube is given by the following equation:

$\text{Surface Area}=6s^2$ , where is the length of a side.

Plugging in the surface area given by the equation, we can then find the side length of the cube.

Now, recall that the volume of a cube is given by the following equation:

$\text{Volume}=s^3$

$\text{Volume}=5^3=125$

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Question

In cubic inches, find the volume of a tetrahedron that has a surface area of $100\sqrt3\textup{ in}^{2}$ .

Answer

First, we will need to find the length of a side of the tetrahedron.

We can use the surface area to find the lengh of a side. Recall that the formula to find the surface of a tetrahedron:

$\text{Surface Area}=\sqrt3 s^2$ , where is the side length.

$100\sqrt3=\sqrt3s^2$

Now, recall the formula to find the volume of a tetrahedron:

$\text{Volume}=\frac{\sqrt2}{12}s^3$

$\text{Volume}=\frac{\sqrt2}{12}10^3=\frac{1000\sqrt2}{12}=\frac{250\sqrt2}{3}$

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Question

Roberto has a swimming pool that is in the shape of a rectangular prism. His swimming pool is meters wide, meters long, and meters deep. He needs to fill up the pool for summer, and his hose fills at a rate of cubic meters per hour. How many hours will it take for Roberto to fill up the swimming pool?

Answer

First, find the volume of the pool. For a rectangular prism, the formula for the volume is the following:

$\text{Volume}=\text{length}\times \text{width}\times \text{height}$

For the swimming pool,

$\text{Volume}=10 \times 25\times 3=750$ cubic meters

Now, because the hose only fills up cubic meters per hour, divide the total volume by to find how long it will take for the pool to fill.

$750\div5=150$

It will take the pool hours to fill by hose.

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Question

A pyramid is placed inside a cube so that they share a base and height. If the surface area of the cube is $486\textup{ft}^{2}$ , what is the volume of the pyramid, in square feet?

Answer

First, we need to find the length of a side for the cube.

Recall that the surface area of the cube is given by the following equation:

$\text{Surface Area}=6s^2$ , where is the length of a side.

Plugging in the surface area given by the equation, we can then find the side length of the cube.

Now, because the pyramid and the cube share a base, we know that the pyramid must be a square pryamid.

Recall how to find the volume of a pyramid:

$\text{Volume}=\text{Area of base}\times \text{height}\times \frac{1}{3}$

$\text{Area of Base}=9\times9=81$

Now, since the pyramid is the same height as the cube, the height of the pyramid is also .

$\text{Volume}=81\times 9\times \frac{1}{3}=243$

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Question

The volume of the right triangular prism is $120\textup{in}^{3}$ . Find the value of .

Answer

The volume of a right triangular prism is given by the following equation:

$\text{Volume}=\text{Area of the Base}\times \text{height}$

Now, for the given question, the height is .

$120=\text{Area of the base}\times 12$

$\text{Area of the base}=10$

Since the area of the base is a right triangle, we can plug in the given values to find .

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

$10=\frac{5x}{2}$

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Question

Matt baked a rectangular cake for his mom's birthday. The cake was inches long, inches wide, and inches high. If he cuts the cake into pieces that are inches long, inches wide, and inches high, how many pieces of cake can he cut?

Answer

First, find the volume of the cake. For a rectangular prism,

$\text{Volume}=\text{length}\times\text{width}\times\text{height}$

$\text{Volume}=15\times12\times4=720$

Next, find the volume of each individual slice.

$\text{Volume}=3\times2\times4=24$

Now, divide the volume of the entire cake by the volume of the slice to get how many pieces of cake Matt can cut.

$720\div24=30$

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Question

The tent shown below is in the shape of a triangular prism. What is the volume of this tent in cubic feet?

Answer

The volume of a right triangular prism is given by the following equation:

$\text{Volume}=\text{Area of the Base}\times \text{height}$

$\text{Area of Base}=\frac{5\times6}{2}=15$

$\text{Volume}=15\times 8=120$

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Question

The height of a box is twice its width and half its length. If the volume of the box is $64\textup{yd}^{3}$ , what is the length of the box?

Answer

For a rectangular prism, the formula for the volume is the following:

$\text{Volume}=\text{length}\times \text{width}\times \text{height}$

Now, we know that the height is twice its width. We can rewrite that as:

$w=\frac{h}{2}$

We also know that the height is half its length. That can be written as:

$h=\frac{1}{2}l$

Now, we can plug in the values of the length, width, and height in terms of height to find the height.

$64=(2h)(\frac{h}{2})(h)$

The question wants to find the length of the box. Plug in the value of the height in the earlier equation we wrote earlier to represent the relatioinship between the height and the length.

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Question

Susan bought a chocolate bar that came in a container shaped like a triangular prism shown below. If the container is completely filled with chocolate, in cubic inches, what volume of chocoate did Susan buy?

Answer

The volume of a right triangular prism is given by the following equation:

$\text{Volume}=\text{Area of the Base}\times \text{height}$

$\text{Area of Base}=\frac{4\times2}{2}=4$

$\text{Volume}=4\times 14=56$

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Question

Troy's company manufactures dice that are shaped like cubes and have side lengths of $2\textup{cm}$ . If the plastic needed to make the dice costs per cubic centimeter, how much does it cost Troy to make one die?

Answer

First, find the volume of the die. For a cube, the volume has the following formula:

$\text{Volume}=\text{side length}^3$

$\text{Volume}=2^3=8$

Because it costs for each cubic centimeter, you will need to multiply this number by to get the cost of each die.

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Question

If the side lengths of a cube are tripled, what effect will it have on the volume?

Answer

Start by taking a cube that is $1\times1\times1$ . The volume of this cube is .

Next, triple the sides of this cube so that it becomes $3\times3\times3$ . The volume of this cube is .

The volume of the new cube is times as large as the original.

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Question

In cubic feet, find the volume of the pentagonal prism illustrated below. The pentagon has an area of $16\textup{ft}^{2}$ , and the prism has a height of $8\textup{ft}$ .

Answer

For any prism, the volume is given by the following equation:

$\text{Volume}=\text{Area of base}\times\text{height}$

The question gives us the area of the base and the height.

$\text{Volume}=16\times8=128$

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