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SAT Math : Geometry

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Example Question #591 : Problem Solving

Aluminum is sold to a soup manufacturer at a rate of $0.0015 per square inch. The cans are made so that the ends perfectly fit on the cylindrical body of the can. It costs $0.00125 to attach the ends to the can. The outer label (not covering the top / bottom) costs $0.0001 per in² to print and stick to the can. The label must be 2 inches longer than circumference of the can. Ignoring any potential waste, what is the manufacturing cost (to the nearest cent) for a can with a radius of 5 inches and a height of 12 inches?

Possible Answers:

$0.45

$0.84

$0.57

$0.77

$0.91

Correct answer:

$0.84

Explanation:

We have the following categories to consider:

<Aluminum Cost> = (<Area of the top and bottom of the can> + <Lateral area of the can>) * 0.0015

<Label Cost> = (<Area of Label>) * 0.0001

<Attachment cost> = 2 * 0.00125 = $0.0025

The area of ends of the can are each equal to π*5² or 25π. For two ends, that is 50π.

The lateral area of the can is equal to the circumference of the top times the height, or 2 * π * r * h = 2 * 5 * 12 * π = 120π.

Therefore, the total surface area of the aluminum can is 120π + 50π = 170π. The cost is 170π * 0.0015 = 0.255π, or approximately $0.80.

The area of the label is NOT the same as the lateral area of the can. (Recall that it must be 2 inches longer than the circumference of the can.) Therefore, the area of the label is (2 + 2 * π * 5) * 12 = (2 + 10π) * 12 = 24 + 120π. Multiply this by 0.0001 to get 0.0024 + 0.012π = (approximately) $0.04.

Therefore, the total cost is approximately 0.80 + 0.04 + 0.0025 = $0.8425, or $0.84.

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Example Question #161 : Solid Geometry

The number of square units in the surface area of a right circular cylinder is equal to the number of cubic units in its volume. If r and h represent the length in units of the cylinder's radius and height, respectively, which of the following is equivalent to r in terms of h?

Possible Answers:

r = 2h/(h – 2)

r = h²/(h + 2)

r = h² + 2h

r = h/(2h – 2)

r = 2h² + 2

Correct answer:

r = 2h/(h – 2)

Explanation:

We need to find expressions for the surface area and the volume of a cylinder. The surface area of the cylinder consists of the sum of the surface areas of the two bases plus the lateral surface area.

surface area of cylinder = surface area of bases + lateral surface area

The bases of the cylinder will be two circles with radius r. Thus, the area of each will be πr², and their combined surface area will be 2πr².

The lateral surface area of the cylinder is equal to the circumference of the circular base multiplied by the height. The circumferece of a circle is 2πr, and the height is h, so the lateral area is 2πrh.

surface area of cylinder = 2πr² + 2πrh

Next, we need to find an expression for the volume. The volume of a cylinder is equal to the product of the height and the area of one of the bases. The area of the base is πr², and the height is h, so the volume of the cylinder is πr²h.

volume = πr²h

Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.

2πr² + 2πrh = πr²h

First, let's factor out 2πr from the left side.

2πr(r + h) = πr²h

We can divide both sides by π.

2r(r + h) = r²h

We can also divide both sides by r, because the radius cannot equal zero.

2(r + h) = rh

Let's now distribute the 2 on the left side.

2r + 2h = rh

Subtract 2r from both sides to get all the r's on one side.

2h = rh – 2r

rh – 2r = 2h

Factor out an r from the left side.

r(h – 2) = 2h

Divide both sides by h – 2

r = 2h/(h – 2)

The answer is r = 2h/(h – 2).

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Example Question #1 : How To Find Surface Area

What is the surface area of a cylinder with a radius of and a height of ?

Possible Answers:

$108\pi$

$96\pi$

$164\pi$

$88\pi$

Correct answer:

$88\pi$

Explanation:

When you're calculating the surface area of a cylinder, note that the cylinder will have two circles, one for the top and one for the bottom, and one rectangle that wraps around the "side" of the cylinder (it's helpful to picture peeling the label off a can of soup - it's curved when it's on the can, but really it's a rectangle that has been wrapped around). You know the area of the circle formula; for the rectangle, note that the height is given to you but the width of the rectangle is one you have to intuit: it's the circumference of the circle, because the entire distance around the circle from one point around and back again is the horizontal distance that the area must cover.

Therefore the surface area of a cylinder = $2\pi r^2+2\pi rh=2\times4^2\times \pi +2\times 4\times 7 \times \pi=32\pi +56\pi =88\pi$

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Example Question #11 : How To Find The Surface Area Of A Cone

What is the surface area of a cone with a radius of 4 and a height of 3?

Possible Answers:

$48\pi$

$25\pi$

$40\pi$

$36\pi$

$16\pi$

Correct answer:

$36\pi$

Explanation:

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

$\Pi r^{2} + \Pi r\sqrt{r^{2} + h^{2}}= \Pi\ast 4^{2} + \Pi \ast 4\sqrt{4^{2} + 3^{2}} = 16\Pi + 4\Pi \sqrt{25} = 16\Pi + 20\Pi = 36\Pi$

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Example Question #1 : How To Find The Surface Area Of A Cone

The lateral area is twice as big as the base area of a cone. If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

Possible Answers:

81π

9π

90π

27π

54π

Correct answer:

81π

Explanation:

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r²)

rl = 2r²

l = 2r

Cone

From the diagram, we can see that r² + h² = l². Since h = 9 and l = 2r, some substitution yields

r² + 9² = (2r)²

r² + 81 = 4r²

81 = 3r²

27 = r²

B = π(r²) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

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Example Question #11 : Solid Geometry

A right cone has a radius of 4R and a height of 3R. What is the ratio of the total surface area of the cone to the surface area of just the base?

Possible Answers:

$\frac{9}{5}$

$\frac{7}{6}$

$\frac{9}{4}$

$\frac{3}{2}$

Correct answer:

$\frac{9}{4}$

Explanation:

We need to find total surface area of the cone and the area of the base.

The area of the base of a cone is equal to the area of a circle. The formula for the area of a circle is given below:

$A=\pi r^2$ , where r is the length of the radius.

In the case of this cone, the radius is equal to 4R, so we must replace r with 4R.

$A=\pi(4R)^2=\pi(4)^2(R)^2=16\pi R^2$

To find the total area of the cone, we need the area of the base and the lateral surface area of the cone. The lateral surface area (LA) of a cone is given by the following formula:

$LA=\frac{1}{2}(2\pi r)(l)$ , where r is the radius and l is the slant height.

We know that r = 4R. What we need now is the slant height, which is the distance from the edge of the base of the cone to the tip.

In order to find the slant height, we need to construct a right triangle with the legs equal to the height and the radius of the cone. The slant height will be the hypotenuse of this triangle. We can use the Pythagorean Theorem to find an expression for l. According to the Pythagorean Theorem, the sum of the squares of the legs (which are 4R and 3R in this case) is equal to the square of the hypotenuse (which is the slant height). According to the Pythagorean Theorem, we can write the following equation:

(4R)^2+(3R)^2=l^2

16R^2+9R^2=l^2

25R^2=l^2

l=5R

Let's go back to the formula for the lateral surface area (LA).

$LA=\frac{1}{2}(2\pi r)(l)$

$LA=\frac{1}{2}(2\pi(4R))(5R)$

$LA=20\pi R^2$

To find the total surface area (TA), we must add the lateral area and the area of the base.

$TA=20\pi R^2+16\pi R^2=36\pi R^2$

The problem requires us to find the ratio of the total surface area to the area of the base. This means we must find the following ratio:

$\frac{36\pi R^2}{16\pi R^2}$

We can cancel $\pi R^2$ , which leaves us with 36/16.

Simplifying 36/16 gives 9/4.

The answer is 9/4.

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Example Question #171 : Solid Geometry

You are given a right circular cone with height $5\ cm$ . The radius is twice the length of the height. What is the volume?

Possible Answers:

$50\pi cm^3$

$100\pi cm^3$

$\frac{50}{3}\pi cm^3$

$500\pi cm^3$

$\frac{500}{3}\pi cm^3$

Correct answer:

$\frac{500}{3}\pi cm^3$

Explanation:

You are given a right circular cone with height 5. The radius is twice the length of the height. What is the volume?

Height = 5cm. The radius is twice the height. Radius=2*Height , so the radius is $10\ cm$ .

$Volume=\pi r^2\frac{h}{3}=\pi(10)^2\frac{5}{3}=\frac{500}{3}\pi cm^3$

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Example Question #2 : How To Find The Surface Area Of A Cone

Cone

In terms of , express the surface area of the above right circular cone.

Possible Answers:

$A= \pi r^{2} + \pi r\sqrt{ 576 - r^{2} }$

None of the other choices gives the correct response.

$A = \pi r^{2} + 24 \pi r$

$A = 24 \pi r$

$A=8 \pi r^{2}$

Correct answer:

$A = \pi r^{2} + 24 \pi r$

Explanation:

The surface area of a right circular cone, given its slant height and the radius of its base, can be found using the formula

$A = \pi r^{2 }+ \pi r l$

The slant height is shown to be 24, so setting l = 24 and substituting:

$A = \pi r^{2 }+ \pi r \cdot 24$

$A = \pi r^{2 }+ 24 \pi r$

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Example Question #4 : How To Find The Surface Area Of A Cone

Cone

In terms of , express the surface area of the provided right circular cone.

Possible Answers:

$A= \pi r^{2}+ \pi r \sqrt{r^{2} + 400}$

$A= \frac{20}{3} \pi r^{2}$

$A= \pi r^{2}+ 20 \pi r$

$A= \frac{1}{3} \pi r^{2} \sqrt{r^{2} + 400}$

$A = \frac{1}{3} \pi r^{2} \sqrt{r^{2} - 400}$

Correct answer:

$A= \pi r^{2}+ \pi r \sqrt{r^{2} + 400}$

Explanation:

The surface area of a right circular cone, given its slant height and the radius of its base, can be found using the formula

$A = \pi r^{2 }+ \pi r l$

The height is shown in the diagram to be 20. By the Pythagorean Theorem,

$l^{2} = r ^{2} + h^{2}$

Setting h = 20 and solving for :

$l^{2} = r ^{2} + 20 ^{2}$

$l^{2} = r ^{2} + 400$

$l =\sqrt{ r ^{2} + 400}$

Substituting in the surface area formula:

$A = \pi r^{2 }+ \pi r\sqrt{ r ^{2} + 400}$

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Example Question #1 : How To Find The Volume Of A Cone

An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?

Possible Answers:

π(r²)(h)/(60w)

π(r²)(h)/(180w)

20w/(π(r²)(h))

π(r²)(h)/(20w)

180w/(π(r²)(h))

Correct answer:

π(r²)(h)/(180w)

Explanation:

The volume of a cone is given by the formula V = (πr²)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr²)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr²)/(3w) * (1/60) = π(r²)(h)/(180w)

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SAT Math : Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #591 : Problem Solving

Example Question #161 : Solid Geometry

Example Question #1 : How To Find Surface Area

Example Question #11 : How To Find The Surface Area Of A Cone

Example Question #1 : How To Find The Surface Area Of A Cone

Example Question #11 : Solid Geometry

Example Question #171 : Solid Geometry

Example Question #2 : How To Find The Surface Area Of A Cone

Example Question #4 : How To Find The Surface Area Of A Cone

Example Question #1 : How To Find The Volume Of A Cone

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