Cones - ACT Math

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Question

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

Answer

Find the slant height of the cone using the Pythagorean theorem: _r_2 + _h_2 = _s_2 resulting in 62 + 82 = _s_2 leading to _s_2 = 100 or s = 10 in

SA = πrs + πr_2 = π(6)(10) + π(6)2 = 60_π + 36_π_ = 96_π_ in2

60_π_ in2 is the area of the cone without the base.

36_π_ in2 is the area of the base only.

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Question

Use the following formula to answer the question.

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

The slant height of a right circular cone is . The radius is , and the height is . Determine the surface area of the cone.

Answer

Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is .

First plug these numbers into the equation provided.

$SA=\pi r^2+\pi rl=\pi (2)^2+\pi (2)(6)=4\pi cm^2+12\pi cm^2$

Then simplify by combining like terms.

$4\pi cm^2+12\pi cm^2=16\pi cm^2$

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Question

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

Answer

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r L$ .

The diameter of the base is $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

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Question

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

Answer

The formula for the surface area of a cone with base of radius and slant height is

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

The diameter of the base is twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

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

$l = \sqrt{h^{2}+ h^{2}}$

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

$l =h \sqrt{2}$

Substitute $h \sqrt{2}$ for and for in the surface area formula:

$A = \pi h^{2} + \pi h \cdot h \sqrt{2}$

$A = \pi h^{2} + \pi h ^{2} \sqrt{2}$

$A = \left (1 + \sqrt{2} \right ) \pi h^{2}$

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Question

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

Answer

The formula for the surface area of a cone with base of radius and slant height is

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

The base has radius and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

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

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

$l = \sqrt{r^{2}+ 16r^{2}}$

$l = \sqrt{17r^{2}}$

$l = r \sqrt{17 }$

Substitute $r \sqrt{17 }$ for in the surface area formula:

$A = \pi r^{2} + \pi r \cdot r \sqrt{17}$

$A = \pi r^{2} + \pi r^{2} \sqrt{17}$

$A = \left ( 1+ \sqrt{17} \right ) \pi r^{2}$

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Question

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

Answer

The formula for the surface area of a cone with base of radius and slant height is

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

The diameter of the base is the circumference divided by $\pi$ , which is

$d = C \div (2\pi ) \approx 100 \div 3.14 \approx 31.83$

This is also the height .

The radius is half this, or

$r = d \div 2 \approx 31.83 \div 2 \approx 15.92$

The slant height can be found by way of the Pythagorean Theorem:

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

$l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}$

$l \approx \sqrt{253.30+ 1,013.21}$

$l \approx \sqrt{1266.51}$

$l \approx 35.60$

Substitute in the surface area formula:

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

$A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60$

$A \approx 795.8+ 1,780.1$

$A \approx 2,575$

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Question

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

Answer

The formula for the surface area of a cone with base of radius and slant height is

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

The slant height is twice the diameter, or, equivalently, four times the radius, so

and

$A = \pi r^{2} + \pi r \cdot 4r$

$A = \pi r^{2} +4 \pi r^{2}$

$A = 5 \pi r^{2}$

The radius of the base is the circumference divided by $2\pi$ , which is

$80 \div 2 \pi \approx 80 \div (2 \times 3.14 )\approx 12.73$

Substitute:

$A = 5 \pi r^{2}$

$A \approx 5 \cdot 3.14 \cdot (12.73)^{2}$

$A \approx 2,546$

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Question

The radius of the base of a cone is ; its slant height is two-thirds of the diameter of that base. Give its surface area in terms of .

Answer

The formula for the surface area of a cone with base of radius and slant height is

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

The diameter of the base is twice radius , or , and its slant height is two-thirds of this diameter, which is $\frac{2}{3}\cdot 2r = \frac{4}{3}r$ . Substitute this for in the formula:

$A = \pi r^{2} + \pi r \cdot \frac{4}{3}r$

$A = \pi r^{2} + \frac{4}{3} \pi r ^{2}$

$A = \frac{7}{3} \pi r ^{2} = \frac{7\pi r ^{2}}{3}$

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Question

Some conical party hats are being decorated with glitter paper. If the base of the hats is inches across and the slant height is inches, how many square inches of glitter paper are needed to decorate one hat?

Answer

The formula for the surface area of a cone is

$\pi r^2 + \pi rl$ , where is the slant height. Since we're only concerned with the slant height portion of the surface area formula, we can ignore the $\pi r^2$ portion of the equation.

Plug in known values to this part of the equation and solve.

$\pi rl = \pi (6)(8) = 48\pi$

So, each hat requires $48\pi \textup{ in}^2$ of glitter paper.

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Question

A heat shield on a particular satellite takes the form of a cone. If the surface area of the "face" of the cone (not counting the disk on the bottom) is , and the stripe of reflective paint from the tip of the cone is down to the base is feet long, what is the diameter of the disk in feet? Round $\pi$ to 3 significant digits. Round your final answer to the nearest foot.

Answer

In this problem, we only need to consider the part of the formula for conic surface area that deals with slant height, since that is all the information we have.

We know the formula for the lateral surface area of a cone is $\pi r l$ , and we know that is feet. Plugging in our other values gives us:

$\pi r (38) = 2744.36$

Simplify:

$\pi r (38) = 2744.36$

$r = \frac{2744.36}{(38)(3.14)} \approx 23$

Thus, if our radius is approximately feet, our diameter is approximately feet.

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Question

What is the surface area in square units of a cone with radius units and height units?

Answer

The formula for surface area of a cone is:

$\pi r^2 + \pi r l$ , where is the slant height. Since we know the radius, we can calculate the first part without issue:

$\pi r^2 = \pi (7)^2 = 49\pi$

The second part requires us to calculate slant height. Since all cones have a right angle created by the base and height perpendicular to the base, we can use the Pythagorean theorem to calculate :

Now, we can complete our formula. Don't forget to add in the circular base.

$\pi r^2 + \pi r l = 49\pi + \pi (7)(25) = 49\pi + 175\pi = 224\pi$

Thus, our surface area is $224\pi$ square units.

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Question

The surface area of a cone equals $24\pi \textup{cm}^2.$ If the radius of the cone equals $3\textup{cm}$ , what is the height of the cone?

Answer

To solve this question, you need to know the surface area of a cone equation: $\textup{surface area}=\pi r(r+\sqrt{h^2 +r^2}),$ where is the radius of the cone $(3\textup{cm})$ , and is the height of the cone. We must substitute all of the values that we know, and solve for the height of the cone to get the answer.

$24\textup{cm}^2=\pi *3\textup{cm}(3\textup{cm}+\sqrt{h^2 +(3\textup{cm})^2}),$ Therefore, the height of the cone is $4\textup{cm.}$

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Question

You have an empty cylinder with a base diameter of 6 and a height of 10 and you have a cone full of water with a base radius of 3 and a height of 10. If you empty the cone of water into the cylinder, how much volume is left empty in the cylinder?

Answer

Cylinder Volume = $\pir^{2}h$

Cone Volume = $\frac{1}{3}\pir^{2}h$

Cylinder Diameter = 6, therefore Cylinder Radius = 3

Cone Radius = 3

Empty Volume = Cylinder Volume – Cone Volume

$=\pir^{2}h-\frac{1}{3}\pir^{2}h$

$=\pi3^{2}10-\frac{1}{3}\pi3^{2}10$

$=\pi90-\pi30$

$=\pi60$

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Question

You have two cones, one with a diameter of and a height of and another with a diameter of and a height of . If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?

Answer

1. Find the volume of each cone:

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

Cone 1:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (3^{2})(10)=30\pi$

Cone 2:

Since the diameter is , the radius is .

$Volume=\frac{1}{3}\pi (1.5^{2})(8)=6\pi$

2. Subtract the smaller volume from the larger volume:

$30\pi-6\pi=24\pi$

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Question

You have an empty cone and a cylinder filled with water. The cone has a diameter of and a height of . The cylinder has a diameter of and a height of . If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?

Answer

1. Find the volumes of the cone and cylinder:

Cone:

Since the diameter is , the radius is .

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

$Volume=\frac{1}{3}\pi (4^{2})(9)=48\pi$

Cylinder:

Since the diameter is , the radius is .

$Volume=\pi r^{2}h$

$Volume=\pi (5^{2})(12)=300\pi$

2. Subtract the cone's volume from the cylinder's volume:

$300\pi-48\pi=252\pi$

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Question

What is the volume of a cone with a height of 7 cm and a radius of 4 cm? Leave your answer in terms of $\pi$ and as a fraction if need be.

Answer

To find the volume of a cone plug the radius and height into the formula for the volume of a cone.

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

$\=\frac{1}{3}\pi 4^2\cdot 7\ \=\frac{112}{3}\pi cm^3$

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Question

What is the volume of a cone with a radius of 3 mm and a height of 6 mm?

Answer

The formula for the volume of a cone is given by the equation:
$V = \frac{1}{3} \pi r^2h$ .

Pluggin in our values we get:

$V = \frac{1}{3} \pi 3^26$

$V=18\pi mm^3$

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Question

Which of the following will quadruple the volume of a cone?

Doubling the radius

Doubling the height

Quadrupling the height

Answer

Bearing in mind the volume formula for a cone:

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

Because the volume varies by the square of the radius, doubling the radius will quadruple the volume (since .) Because the volume also varies linearly by the height, quadrupling the height will quadruple the volume.

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Question

What is the volume of a cone with a radius of and a height of ? Leave your answer in terms of $\pi$ , reduce all fractions.

Answer

To find the volume of a cone with radius , and height use the formula:

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

We plug in our given radius and height to find:

$V = \frac{1}{3}\pi * 6^2*2 = \frac{72}{3}\pi = 24\pi$

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Question

A conical paper cup is being used as a makeshift funnel for motor oil. If the cup is 100 millimeters deep at the center and has a radius of 70 millimeters, how many cubic millimeters of motor oil can it hold at one time? Round your final answer to the nearest integer.

Answer

The formula for the volume of a cone is:

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

With the information we have, we can simply plug values into this equation and solve.

$V = \frac{\pi r^2h}{3} = \frac{(100^2)(70)\pi}{3} \approx 233333\pi$

So our cone can hold approximately $233333\pi$ cubic millimeters of motor oil.

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