Geometry - Math

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Question

Find the surface area of the following half-cylinder.

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Answer

The formula for the surface area of a half-cylinder must include one-half of the surface area of a cylinder, which would be:

$SA = \frac{1}{2} (lateral\ area + 2 \cdot (base))$

We also need to add the area of the new rectangular face that is created by cutting the cylinder in half. The area of this rectangle would be:

$A=l\times w$

where the length of the rectangle is the same as the height of the half-cylinder, and the width of the rectangle is the same as the diameter of the base of the half-cylinder. So we can rewrite the area of the rectangle as:

$A=h\p\times d$

Now we can combine the two area formulas to find the total surface area of the half-cylinder:

$SA = \frac{1}{2} (2 \pi r (h) + 2 (\pi r^2))+h\cdot d$

$SA = \pi r (h) + \pi r^2+hd$

where is the radius of the base and is the length of the height, and is the diameter of the base.

Plugging in our values, we get:

$SA = \pi (5m) (15m) + \pi (5m)^2+15\cdot 10$

$SA = 75 \pi m^2 + 25 \pi m^2+150 = 150+100 \pi m^2$

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Question

Find the surface area of the following polyhedron.

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Answer

The formula for the surface area of the polyhedron is:

$SA = SA_{cone}+\frac{1}{2}SA_{sphere}$

$SA = \pi r l+\frac{1}{2}(4 \pi r^2)$

$SA = \pi r l+2 \pi r^2$

Where is the radius of the cone, is the slant height of the cone, and is the radius of the sphere

Use the formula for a triangle to find the radius and slant height:

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

$3\sqrt{3}m-9m-6\sqrt{3}m$

Plugging in our values, we get:

$SA = \pi r l+2 \pi r^2$

$SA = (\pi)(3\sqrt{3}m)(6\sqrt{3}m)+2(\pi)(3\sqrt{3}m)^2$

$SA = 54 \pi m^2 + 54 \pi m^2 = 108 \pi m^2$

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Question

Find the surface area of the following polyhedron.

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Answer

The formula for the surface area of the polyhedron is:

$SA = (cone:no\ base)+(cylinder:one\ base)$

$SA = \frac{1}{2}(circumference)(slant\ height)+(base)+(circumference)(height)$

$SA = \frac{1}{2}(2 \pi r)(h_s)+(\pi r^2)+(2 \pi r)(h)$

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

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

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

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

Plugging in our values, we get:

$SA = \frac{1}{2}(2\pi (3m))(6m)+(\pi)(3m)^2+(2\pi (3m)(9m))$

$SA = 81 \pi m^2$

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Question

Find the surface area of the following polyhedron.

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Answer

The formula for the surface area of a polyhedron is:

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

$SA = \frac{1}{2}(2\pi r)(h_s)+ \frac{1}{2}(4\pi r^2)$

$SA = \pi rh_s+ 2\pi r^2$

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

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

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

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

Plugging in our values, we get:

$SA = \pi (4\sqrt{2}m)(8m)+ 2\pi (4\sqrt{2}m)^2$

$SA = 32\sqrt{2}\pi + 64 \pi m^2$

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Question

Find the volume of the following half cylinder.

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Answer

The formula for the volume of a half-cylinder is:

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

where is the radius of the base and is the length of the height.

Plugging in our values, we get:

$V = \frac{1}{2} \pi (5m)^2 (15m)$

$V = 187.5 \pi m^3$

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Question

Find the volume of the following polyhedron.

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Answer

The formula for the volume of the polyhedron is:

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

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

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

Where is the radius of the cone, is the height of the cone, and is the radius of the sphere.

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

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

$3\sqrt{3}m-9m-6\sqrt{3}m$

Plugging in our values, we get:

$V = \frac{\pi (3\sqrt{3}m)^29m}{3}+\frac{2 \pi (3\sqrt{3}m)^3}{3}$

$V = \frac{\pi (27m^2)9m}{3}+\frac{2 \pi (81\sqrt{3}m^3)}{3}$

$V = 81 \pi m^3+54\sqrt{3} \pi m^3$

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Question

Find the volume of the following polyhedron.

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Answer

The formula for the volume of the polyhedron is:

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

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 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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Question

Find the volume of the following polyhedron.

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Answer

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 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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Question

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

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Answer

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 ():

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

$\rightarrow 8 + 8 = c^2$

$\rightarrow 16 = c^2$

$\rightarrow c = 4$

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.

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

$\rightarrow 8 + 16 = c^2$

$\rightarrow 24 = c^2$

$\rightarrow \sqrt{24} = c$

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

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Question

Find the length of the diagonal of the following cube:

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Answer

To find the length of the diagonal, use the formula for a 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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Question

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?

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Answer

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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Question

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

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Answer

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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Question

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

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Answer

The standard equation for surface area is

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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Question

Find the length of an edge of the following cube:

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

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Answer

The formula for the volume of a cube is

,

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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Question

Find the length of an edge of the following cube:

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

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Answer

The formula for the volume of a cube is

,

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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Question

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

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Answer

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

$SA=6\times l^2$

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

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

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Question

If the surface area of a cube equals 96, what is the length of one side of the cube?

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Answer

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a2 → a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

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Question

A sphere with a volume of $\frac{32}{3}$ $\pi m^{3}$ is inscribed in a cube, as shown in the diagram below.

What is the surface area of the cube, in $m^{2}$ ?

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Answer

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

$V_{sphere}=\frac{4}{3}\pi r^{3}$

$\frac{32}{3}\pi =\frac{4}{3}\pi r^{3}$

$\frac{32}{3}=\frac{4}{3}r^{3}$

$\frac{3}{4}\cdot \frac{32}{3}= r^{3}$

$8=r^{3}$

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

$side=2\cdot 2=4$

The formula for the surface area of a cube is:

$SA_{cube}=6s^{2}$

$6s^{2}=6\cdot (4)^{2}=6\cdot 16=96$

The surface area of the cube is $96\ m^{2}$

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Question

What is the surface area of a cube if its height is 3 cm?

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Answer

The area of one face is given by the length of a side squared.

$A_{face}=(3cm)^2=9\ cm^2$

The area of 6 faces is then given by six times the area of one face: 54 cm2.

$A_{total}=6(A_{face})=6(9\ cm^2)=54\ cm^2$

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Question

The side of a cube has a length of $\dpi{100} \small 5 cm$ . What is the total surface area of the cube?

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Answer

A cube has 6 faces. The area of each face is found by squaring the length of the side.

$\dpi{100} \small 5\times 5 = 25$

Multiply the area of one face by the number of faces to get the total surface area of the cube.

$\dpi{100} \small 25 \times 6=150$

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