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SAT Math : Solid Geometry

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Example Question #171 : 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:

$\frac{9}{4}$

$\frac{7}{6}$

$\frac{9}{5}$

$\frac{3}{2}$

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

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:

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

$50\pi cm^3$

$500\pi cm^3$

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

$100\pi cm^3$

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 #3 : 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:

None of the other choices gives the correct response.

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

$A = 24 \pi r$

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

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

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 #1 : Cones

Cone

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

Possible Answers:

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

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

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

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

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

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 : Cones

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)/(180w)

π(r²)(h)/(60w)

180w/(π(r²)(h))

π(r²)(h)/(20w)

20w/(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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Example Question #2 : How To Find The Volume Of A Cone

A cone has a base radius of 13 in and a height of 6 in. What is its volume?

Possible Answers:

None of the other answers

1352π in³

338π in³

4394π in³

1014π in³

Correct answer:

338π in³

Explanation:

The basic form for the volume of a cone is:

V = (1/3)πr²h

For this simple problem, we merely need to plug in our values:

V = (1/3)π13²* 6 = 169 * 2π = 338π in³

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Example Question #872 : Sat Mathematics

A cone has a base circumference of 77π in and a height of 2 ft. What is its approximate volume?

Possible Answers:

8893.5π in³

11,858π in³

71,148π in³

2964.5π in³

142,296π in³

Correct answer:

11,858π in³

Explanation:

There are two things to be careful with here. First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2πr, or, for our values 77π = 2πr. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr²h

For our values this would be:

V = (1/3)π * 38.5² * 24 = 8 * 1482.25π = 11,858π in³

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Example Question #3 : Cones

What is the volume of a right cone with a diameter of 6 cm and a height of 5 cm?

Possible Answers:

$45\pi \ cm^{3}$

$120\pi \ cm^{3}$

$180\pi \ cm^{3}$

$60\pi \ cm^{3}$

$15\pi \ cm^{3}$

Correct answer:

$15\pi \ cm^{3}$

Explanation:

The general formula is given by $V = 1/3Bh = 1/3\pi r^{2}h$ , where = radius and = height.

The diameter is 6 cm, so the radius is 3 cm.

$V=\frac{\pi 3^2\times 5}{3}=15\pi$

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

There is a large cone with a radius of 4 meters and height of 18 meters. You can fill the cone with water at a rate of 3 cubic meters every 25 seconds. How long will it take you to fill the cone?

Possible Answers:

$600\ \textup{minutes}$

$800\ \textup{minutes}$

$3\ \textup{minutes}\ 20\ \textup{seconds}$

$41\ \textup{minutes}\ 53\ \textup{seconds}$

$60\ \textup{minutes}$

Correct answer:

$41\ \textup{minutes}\ 53\ \textup{seconds}$

Explanation:

First we will calculate the volume of the cone

$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

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

Find the volume of a cone with a radius of and a height of .

Possible Answers:

$9000\pi$

$900\pi$

$3000\pi$

$300\pi$

$6000\pi$

Correct answer:

$3000\pi$

Explanation:

Write the formula to find the volume of a cone.

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

Substitute the known values and simplify.

$V=\frac{1}{3}\pi (10)^2 (90)= \frac{1}{3}\pi (100)(90)= \pi (100)(30)=3000\pi$

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SAT Math : Solid Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #171 : Solid Geometry

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

Example Question #3 : How To Find The Surface Area Of A Cone

Example Question #1 : Cones

Example Question #1 : Cones

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

Example Question #872 : Sat Mathematics

Example Question #3 : Cones

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

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

All SAT Math Resources

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