Cylinders - PSAT Math
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An upright cylinder with a height of 30 and a radius of 5 is in a big tub being filled with oil. If only the top 10% of the cylinder is visible, what is the surface area of the submerged cylinder?
An upright cylinder with a height of 30 and a radius of 5 is in a big tub being filled with oil. If only the top 10% of the cylinder is visible, what is the surface area of the submerged cylinder?
The height of the submerged part of the cylinder is 27cm. 2πrh + πr2 is equal to 270π + 25π = 295π
The height of the submerged part of the cylinder is 27cm. 2πrh + πr2 is equal to 270π + 25π = 295π
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A right circular cylinder has a height of 41 in. and a lateral area (excluding top and bottom) 512.5π in2. What is the area of its bases?
A right circular cylinder has a height of 41 in. and a lateral area (excluding top and bottom) 512.5π in2. What is the area of its bases?
The lateral area (not including its bases) is equal to the circumference of the base times the height of the cylinder. Think of it like a label that is wrapped around a soup can. Therefore, we can write this area as:
A = h * π * d or A = h * π * 2r = 2πrh
Now, substituting in our values, we get:
512.5π = 2 * 41*rπ; 512.5π = 82rπ
Solve for r by dividing both sides by 82π:
6.25 = r
From here, we can calculate the area of a base:
A = 6.252π = 39.0625π
NOTE: The question asks for the area of the bases. Therefore, the answer is 2 * 39.0625π or 78.125π in2.
The lateral area (not including its bases) is equal to the circumference of the base times the height of the cylinder. Think of it like a label that is wrapped around a soup can. Therefore, we can write this area as:
A = h * π * d or A = h * π * 2r = 2πrh
Now, substituting in our values, we get:
512.5π = 2 * 41*rπ; 512.5π = 82rπ
Solve for r by dividing both sides by 82π:
6.25 = r
From here, we can calculate the area of a base:
A = 6.252π = 39.0625π
NOTE: The question asks for the area of the bases. Therefore, the answer is 2 * 39.0625π or 78.125π in2.
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The diameter of the lid of a right cylindrical soup can is 5 in. If the can is 12 inches tall and the label costs $0.00125 per square inch to print, what is the cost to produce a label for a can? (Round to the nearest cent.)
The diameter of the lid of a right cylindrical soup can is 5 in. If the can is 12 inches tall and the label costs $0.00125 per square inch to print, what is the cost to produce a label for a can? (Round to the nearest cent.)
The general mechanics of this problem are simple. The lateral area of a right cylinder (excluding its top and bottom) is equal to the circumference of the top times the height of the cylinder. Therefore, the area of this can's surface is: 5π * 12 or 60π. If the cost per square inch is $0.00125, a single label will cost 0.00125 * 60π or $0.075π or approximately $0.24.
The general mechanics of this problem are simple. The lateral area of a right cylinder (excluding its top and bottom) is equal to the circumference of the top times the height of the cylinder. Therefore, the area of this can's surface is: 5π * 12 or 60π. If the cost per square inch is $0.00125, a single label will cost 0.00125 * 60π or $0.075π or approximately $0.24.
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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?
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?
We have the following categories to consider:
= ( + ) * 0.0015
= 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.
We have the following categories to consider:
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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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?
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?
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 _πr_2, and their combined surface area will be 2_πr_2.
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 2_πr_, and the height is h, so the lateral area is 2_πrh_.
surface area of cylinder = 2_πr_2 + 2_πrh_
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 _πr_2, and the height is h, so the volume of the cylinder is πr_2_h.
volume = πr_2_h
Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.
2_πr_2 + 2_πrh_ = πr_2_h
First, let's factor out 2_πr_ from the left side.
2_πr_(r + h) = πr_2_h
We can divide both sides by π.
2_r_(r + h) = r_2_h
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.
2_r_ + 2_h_ = rh
Subtract 2_r_ from both sides to get all the r's on one side.
2_h_ = rh – 2_r_
r_h – 2_r = 2_h_
Factor out an r from the left side.
r(h – 2) = 2_h_
Divide both sides by h – 2
r = 2_h_/(h – 2)
The answer is r = 2_h_/(h – 2).
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 _πr_2, and their combined surface area will be 2_πr_2.
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 2_πr_, and the height is h, so the lateral area is 2_πrh_.
surface area of cylinder = 2_πr_2 + 2_πrh_
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 _πr_2, and the height is h, so the volume of the cylinder is πr_2_h.
volume = πr_2_h
Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.
2_πr_2 + 2_πrh_ = πr_2_h
First, let's factor out 2_πr_ from the left side.
2_πr_(r + h) = πr_2_h
We can divide both sides by π.
2_r_(r + h) = r_2_h
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.
2_r_ + 2_h_ = rh
Subtract 2_r_ from both sides to get all the r's on one side.
2_h_ = rh – 2_r_
r_h – 2_r = 2_h_
Factor out an r from the left side.
r(h – 2) = 2_h_
Divide both sides by h – 2
r = 2_h_/(h – 2)
The answer is r = 2_h_/(h – 2).
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What is the surface area of a cylinder with a radius of 
and a height of 
?
What is the surface area of a cylinder with a radius of 
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 = 
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 =
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A circular swimming pool has diameter 
meters and depth 5 meters throughout. Which of the following expressions gives the total surface area of the inside of the pool, in square meters?
A circular swimming pool has diameter
The pool can be seen as a cylinder with depth (or height) 
, and a base with diameter 
- and, subsequently, radius half this, or 
.
The swimming pool has one circular base, whose area is
The lateral side of the pool has area
Add these:
The pool can be seen as a cylinder with depth (or height)
The swimming pool has one circular base, whose area is
The lateral side of the pool has area
Add these:
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What is the lateral surface area of a cylinder with a base radius of 
and a height of 
?
What is the lateral surface area of a cylinder with a base radius of
First, solve for circumference
The lateral surface area is equal to the circumference times the height. This gives us
First, solve for circumference
The lateral surface area is equal to the circumference times the height. This gives us
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Consider a right, circular cylinder. The circular bases each have an area of 
. If the height of the cylinder is 
, then what is the lateral surface area of the cylinder?
Consider a right, circular cylinder. The circular bases each have an area of
The lateral surface area of a right circular cylinder is equal to the circumference of the base multiplied by the height.
If the base is a circle with area 
, then the radius is 
, so
Therefore, circumference times height gives us
.
The lateral surface area of a right circular cylinder is equal to the circumference of the base multiplied by the height.
If the base is a circle with area
Therefore, circumference times height gives us
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The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:
V = π(d/2)2h = πd2h/4
Sub in h and V: 36p = πd2(4)/4 so 36p = πd2
Thus d = 6
Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:
V = π(d/2)2h = πd2h/4
Sub in h and V: 36p = πd2(4)/4 so 36p = πd2
Thus d = 6
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Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?
Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?
Volume of cylinder = π * (base radius)2 x height = π * (base diameter / 2 )2 x height
Volume Cylinder 1 = π * (3 / 2 )2 x 9 = π * (1.5 )2 x 9 = π * 20.25
Volume Cylinder 2 = π * (4 / 2 )2 x 9 = π * (2 )2 x 9 = π * 36
Total Volume = π * 20.25 + π * 36
Volume of Cylinder 3 = π * (6 / 2 )2 x H = π * (3 )2 x H = π * 9 x H
Set Total Volume equal to the Volume of Cylinder 3 and solve for H
π * 20.25 + π * 36 = π * 9 x H
20.25 + 36 = 9 x H
H = (20.25 + 36) / 9 = 6.25”
Volume of cylinder = π * (base radius)2 x height = π * (base diameter / 2 )2 x height
Volume Cylinder 1 = π * (3 / 2 )2 x 9 = π * (1.5 )2 x 9 = π * 20.25
Volume Cylinder 2 = π * (4 / 2 )2 x 9 = π * (2 )2 x 9 = π * 36
Total Volume = π * 20.25 + π * 36
Volume of Cylinder 3 = π * (6 / 2 )2 x H = π * (3 )2 x H = π * 9 x H
Set Total Volume equal to the Volume of Cylinder 3 and solve for H
π * 20.25 + π * 36 = π * 9 x H
20.25 + 36 = 9 x H
H = (20.25 + 36) / 9 = 6.25”
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A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
LA = 2π(r)(h) = 2π(3)(5) = 30π
LA = 2π(r)(h) = 2π(3)(5) = 30π
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A cylinder has a volume of 20. If the radius doubles, what is the new volume?
A cylinder has a volume of 20. If the radius doubles, what is the new volume?
The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.
The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.
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A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?
A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?
Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.
We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,
2πrh = 54π
Now we substitute 3r in for h.
2πr(3r) = 54π
6πr2 = 54π
Divide by 6π
r2 = 9.
Take the square root.
r = 3.
h = 3r = 3(3) = 9.
Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr2h.
V = πr2h
V = π(3)2(9) = 81π
The answer is 81π.
Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.
We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,
2πrh = 54π
Now we substitute 3r in for h.
2πr(3r) = 54π
6πr2 = 54π
Divide by 6π
r2 = 9.
Take the square root.
r = 3.
h = 3r = 3(3) = 9.
Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr2h.
V = πr2h
V = π(3)2(9) = 81π
The answer is 81π.
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What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?
What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?
The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πrout2 h – πrin2 h. The area of the cylinder using the outer radius is 80π cm3, and resulting hole is given by the volume from the inner radius, 20π cm3. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm3.
The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πrout2 h – πrin2 h. The area of the cylinder using the outer radius is 80π cm3, and resulting hole is given by the volume from the inner radius, 20π cm3. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm3.
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A metal cylindrical brick has a height of 
. The area of the top is 
. A circular hole with a radius of 
is centered and drilled half-way down the brick. What is the volume of the resulting shape?
A metal cylindrical brick has a height of
To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.
The volume of a cylinder is given by the product of the base area times the height: 
.
Find the initial volume using the given base area and height.
Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.
Finally, subtract the volume of the hole from the total initial volume.
To find the final volume, we will need to subtract the volume of the hole from the total initial volume of the cylinder.
The volume of a cylinder is given by the product of the base area times the height:
Find the initial volume using the given base area and height.
Next, find the volume of the hole that was drilled. The base area of this cylinder can be calculated from the radius of the hole. Remember that the height of the hole is only half the height of the block.
Finally, subtract the volume of the hole from the total initial volume.
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What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
What is the volume of a right cylinder with a circumference of 25π in and a height of 41.3 in?
The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr2. Therefore, the total formula for the volume of the cylinder is: V = πr2h.
First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.
Based on this, we know that the volume of our cylinder must be: π*12.52*41.3 = 6453.125π in3
The formula for the volume of a right cylinder is: V = A * h, where A is the area of the base, or πr2. Therefore, the total formula for the volume of the cylinder is: V = πr2h.
First, we must solve for r by using the formula for a circumference (c = 2πr): 25π = 2πr; r = 12.5.
Based on this, we know that the volume of our cylinder must be: π*12.52*41.3 = 6453.125π in3
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What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
What is the volume of a cylinder with a diameter of 13 inches and a height of 27.5 inches?
The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.
Thus, the volume can also be expressed as V = πr2h.
The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.
Now we can easily calculate the volume:
V = 6.52π * 27.5 = 1161.88π in3
The equation for the volume of a cylinder is V = Ah, where A is the area of the base and h is the height.
Thus, the volume can also be expressed as V = πr2h.
The diameter is 13 inches, so the radius is 13/2 = 6.5 inches.
Now we can easily calculate the volume:
V = 6.52π * 27.5 = 1161.88π in3
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An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
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An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in3, what is the mass of the remaining wood after the cylinder is drilled out?
An 12-inch cube of wood has a cylinder drilled out of it. The cylinder has a radius of 3.75 inches. If the density of the wood is 4 g/in3, what is the mass of the remaining wood after the cylinder is drilled out?
We must calculate our two volumes and subtract them. Following this, we will multiply by the density.
The volume of the cube is very simple: 12 * 12 * 12, or 1728 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 3.752 * 12 = 168.75π in3.
The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in3. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.
We must calculate our two volumes and subtract them. Following this, we will multiply by the density.
The volume of the cube is very simple: 12 * 12 * 12, or 1728 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 3.752 * 12 = 168.75π in3.
The volume remaining in the cube after the drilling is: 1728 – 168.75π, or approximately 1728 – 530.1433125 = 1197.8566875 in3. Now, multiply this by 4 to get the mass: (approx.) 4791.43 g.
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