Test your knowledge with 13 solved cylinder exercises. Get ready for ENEM and entrance exams with the questions discussed and ask your questions.
Exercise 1
Calculate the volume of the cylinder and mark the alternative that most closely matches the result. Given: Consider π = 3.14.
a) Volume = 6000 cm³.
b) Volume = 5000 cm³.
c) Volume = 4000 cm³.
d) Volume = 3000 cm³.
e) Volume = 2000 cm³.
Answer: b) Volume = 5000 cm³.
Resolution
To determine the volume of the cylinder, we multiply the base area by the height.
From the image we have the diameter = 16 cm and the height = 25 cm.
Data:
π = 3,14
Radius of the base: r = d/2 = 16/2 = 8 cm
Height: h = 25 cm
Since the base is a circle, its area is obtained by the equation:
A = π.r²
A = π. 8²
A = 3.14. 64
A = 200.96 cm²
The volume of the cylinder is given by the multiplication between the base area and its height.
V = π.r².h
V = 200.96 x 25 = 5024 cm³
Therefore, the volume is about 5000 cm³. This is the best approximation.
Exercise 2
In the Total Cleaning Jet Wash, there was a great movement today, having received 23 customers for complete washing. However, as I started to wash the next car, the water ran out. Only then did the employees realize that the water supplier issued an alert saying that due to repairs and maintenance work, on that day, there would be no supply.
The owner of the establishment asked for an emergency supply with a water truck and the water supplier asked the capacity of the reservoir. As it was very old, the capacity indications had disappeared, being necessary to make the calculation based on its measurements.
The reservoir is a cylinder 4 m high and 1.80 m in diameter.
The water supplier company has five water truck delivery options. Check the option that can be requested by the owner of the lava act, filling your reservoir as much as possible.
Data: 1 m³ = 1 000 l
a) 12 000 l.
b) 11 000 l.
c) 10,000 l.
d) 9,000 l.
e) 8 000 l.
Correct answer: c) 10 000 l.
Resolution
We must calculate the volume in cubic meters and then transform it into liters.
The volume of a cylinder can be calculated by multiplying the base area by the height.
Idea 1: Calculate the base area.
Base area: A = π.r²
Since the diameter is 1.80 m, the radius is 0.90 m.
A = 3.14 * 0.90²
A = 3.14 * 0.81
H = 2.5434 cm²
Idea 2: calculate the volume of cylinder
V = base area * height
V = π.r².h
V = 2.5434 x 4 = 10.1736 m³
IIdea 3: transform the result from m³ to liters.
Since 1 m³ = 1000 l, just multiply the result by 1000.
V = 10.1736 m³ x 1000 = 10 173.6 l
In this way, the owner of the car wash can order the water truck with 10 000 l.
Exercise 3
What is the lateral area of a straight cylinder that is 502.4 cm³ in volume and 8 cm in diameter?
Given: π = 3.14.
a) 355.10 cm²
b) 251.20 cm²
c) 125.51 cm²
d) 375.30 cm²
e) 91.45 cm²
Correct answer: 251.20 cm²
Resolution
Idea 1: Determine height h and radius r
If the diameter is 8 cm, then the radius is equal to 4 cm.
From the volume formula we can determine the height.
Idea 2: lateral area formula
The lateral area of a straight cylinder of height h is equal to that of a rectangle of length equal to the length of the circumference of the cylinder multiplied by the height.
The length of a circle is obtained by the formula: 2.π.r
The area of a rectangle can be obtained by multiplying the base and height values.
Idea 3: perform the calculation of the lateral area
Cylinder side area = 2.π.r.h
Side area = 2. 3,14. 4. 10
Side area = 251.20 cm²
Therefore, the lateral area of the cylinder is 251.20 cm².
Exercise 4
A silo that stores soybeans on a farm had a problem in its structure and needs to be repaired with a solder on the wall. The silo is a tower in the form of a cylinder 10 m high and 6 m in diameter. To carry out the service, the manager decided to empty the silo, temporarily storing the production in buckets of carts in the form of parallelepipeds, with measures equal to 12 m in length, 2 m in width and 1.5 m in height. How many buckets are needed to store all the content?
Use π = 3.14.
a) 15 buckets
b) 7 buckets
c) 16 buckets
d) 9 buckets
e) 8 buckets
Correct answer: e) 8 buckets
Resolution
Idea 1: number of buckets
The number of buckets is the volume of the silo divided by the volume of a truck.
Idea 2: silo volume
As the silo is a cylinder, its volume is obtained by the product between the base area and its height.
base area
Where r is the radius, equal to half the diameter.
A = π.r²
A = 3.14. 3²
A = 28.26 m²
Silo volume
V = A. H
V = π.r².h
V = 28.26. 10
V = 282.60 m³
Idea 3: bucket volume
The bucket volume is the volume of the parallelepiped.
V = length x width x height
V = 12 x 2 x 1.5 = 36 m³
Idea 4: Calculating the number of buckets
Number of buckets = silo volume / bucket volume
Number of buckets = 282.60 / 36
Number of buckets = 7.85
Conclusion
8 buckets will be needed to store the grain.
Exercise 5
(Enem 2010). Dona Maria, a day laborer at the Teixeira family house, needs to make coffee to serve the twenty people who meet at a meeting in the living room. To make coffee, Dona Maria has a cylindrical milk jug and small plastic cups, also cylindrical.
In order not to waste coffee, the daily maid wants to put the minimum amount of water in the milk jug to fill the twenty cups halfway. For this to happen, Dona Maria must
a) fill the milk jug halfway, as it has a volume 20 times greater than the volume of the cup.
b) fill the entire milk jug with water, as it has a volume 20 times greater than the volume of the cup.
c) fill the entire milk jug with water, as it has a volume 10 times greater than the volume of the cup.
d) fill two milk bottles with water, as it has a volume 10 times greater than the volume of the glass.
e) fill five milk bottles with water, as it has a volume 10 times greater than the volume of the glass.
Correct answer: a) fill the milk jug halfway, as it has a volume 20 times greater than the volume of the cup.
Resolution
Idea 1: volume of the cup
As it is a cylinder, the volume is given by:
V = π.r².h
V = π.2².4 = 16π cm²
Idea 2: dairy volume
V = π.r².H
V = π.4².20
V = π. 16. 20 cm³ or 16π x 20 cm³
Here we notice that the dairy has 20 times the volume of 1 cup.
Milk volume = 20 cups volume
Conclusion:
As each cup will be half-filled, the milk jug should be half-filled.
Exercise 6
(Enem 2014) A company that organizes graduation events makes diploma straws from square sheets of paper. So that all the straws are identical, each sheet is wrapped around a wooden cylinder of diameter d in centimeters, without any gap, making 5 complete turns around this cylinder. At the end, a string is tied in the middle of the diploma, well adjusted, so that it does not unfold, as illustrated in the figure.
Then, remove the wooden cylinder from the middle of the rolled paper, finishing the preparation of the diploma. Please consider that the thickness of the original sheet of paper is negligible.
What is the measurement, in centimeters, on the side of the sheet of paper used to make the diploma?
a) πd
b) 2 nd
c) 4 πd
d) 5 πd
e) 10 πd
Correct answer: d) 5 πd
Resolution
Since the paper has been rolled up 5 times, the length of the sheet is equal to 5 times the length of the cylinder circumference.
Length of a circle is given by the formula:
2. π. r
The radius is half the diameter
r = d/2
Substituting in formula
Sheet length = 5. 2. π. d/2
Sheet length = 5πd
Exercise 7
(Enem 2015) The pluviometric index is used to measure the precipitation of rainwater, in millimeters, in a determined period of time. Its calculation is made according to the level of rainwater accumulated in 1 m², that is, if the index is 10 mm, means that the height of the water level accumulated in an open cube-shaped tank with 1 m² of base area, is 10 mm. In one region, after a strong storm, it was found that the amount of rain accumulated in a cylindrical can, with a radius of 300 mm and a height of 1200 mm, was one third of its capacity.
Use 3.0 as an approximation for π.
The pluviometric index of the region, during the storm period, in millimeters, is of
a) 10.8.
b) 12.0.
c) 32.4.
d) 108.0.
e) 324.0.
Correct answer: d) 108.0.
Idea 1: volume of liquid in the can
Using:
π = 3,0
r = 300 mm
h = 1200 mm
Idea 2: Dumping this content into a cube
The cube must have 1 m³ that is,
cube volume = height x width x length
1 m³ = height x 1 m x 1 m
Changing to millimeters and equaling the calculated liquid volume:
108,000,000 mm³ = height x 1000 m x 1000 m
height = 108 mm
Conclusion:
Thus, the measured rainfall index was 108 mm.
Exercise 8
(Enem 2015). A Brazilian fish export factory sells canned tuna abroad, in two types of cans cylindrical: one with a height of 4 cm and radius 6 cm, and another of unknown height and radius of 3 cm, respectively, as shown in the figure. It is known that the measure of the volume of the can that has the largest radius, V1, is 1.6 times the measure of the volume of the can that has the smallest radius, V2.
The unknown height measurement holds
a) 8 cm.
b) 10 cm.
c) 16 cm.
d) 20 cm.
e) 40 cm.
Correct answer: b) 10 cm.
Resolution
V1 = 1.6 V2
Replacing the cylinder volume formulas in V1 and V2:
Since π appears on both sides by multiplying, it can be canceled out. Isolating x, we have:
Therefore, the height x of the tallest can is 10 cm.
Exercise 9
(Enem 2021) A building supply store sells two types of water tanks: type A and type B. Both are cylindrical in shape and have the same volume, and the height of the type B water tank is equal to 25% of the height of the type A water tank.
If R denotes the radius of the type A water tank, then the radius of the type B water tank is
a) R/2
b) 2 R
c) 4 R
d) 5 R
e) 16 R
Correct answer: b) 2 R
Resolution
The statement says that the height of B is 25% of the height of A. That is, A is four times higher than B.
Where a, the height of box A and, b, the height of box B:
a = 4b
Where R is the radius of box type A and r is the measure of the radius of box B.
How the volumes are equal:
VA = VB
Substituting the formulas for the volume of the cylinders, which is π.r².h, we have:
Eliminating the equal terms, which multiply on both sides of the equation:
Applying a square root to both sides of the equation. ("throwing the exponent 2 of r to the left in the form of a square root")
This means that the radius of the type B water tank is twice the length of the radius of the A water tank.
Exercise 10
(Fuvest). A cubic-shaped water tank measuring 1 meter on a side is coupled with a cylindrical pipe measuring 4 cm in diameter and 50 m in length. At one point, the box is full of water and the pipe is empty.
Drop the water through the pipe until it is full. What is the approximate height of the water in the box at the time the pipe was filled?
a) 90 cm.
b) 92 cm.
c) 94 cm.
d) 96 cm.
e) 98 cm.
Correct answer: c) 94 cm
Resolution
Idea 1: pipe volume
Being a cylinder, the volume is given by the product between the area of its base and its height.
Volume = π.r².h
In this case, its height will be the length of 50 m and the base area equal to a section of the pipe.
The radius is half the diameter, and in meters, r = 0.02 m.
Pipe volume = π.0.02². 50 = 0,02π
Idea 2: box volume
Being a cube, height = width = length
The statement says that the measure of the sides of the cube is equal to 1 m.
Box volume = 1 m³
Height x width x length = 1 x 1 x 1 = 1 m³
Idea 3: New height
As the amount of water in the pipe came out of the box, let's do the subtraction:
New water volume in the box = Box volume - Pipe volume
As water leaves the box, the only dimension that can change is height. The width and length remain equal to 1 m.
Height x 1 x 1 = 1 - 0.02π
Making π = 3
Height = 1 - 0.06
Height = 0.94
Conclusion
The new height (water level in the box) is 0.94 m.
Note: Even considering π = 3.1415, the closest result is still the letter c. Look:
Height = 1 - 0.02 x 3.1415
Height = 1 - 0.06283
Height = 0.93717 (approximately 0.94)
Exercise 11
(Cesgranrio) A container in the shape of a straight cylinder, whose base diameter measures 40 cm and height 100/π cm, holds a certain liquid, which occupies 40% of its capacity. The volume of liquid contained in this container is, in liters, approximately equal to:
a) 16
b) 18
c) 20
d) 30
e) 40
Correct answer: a) 16
Resolution
Idea 1: cylinder volume
V = π.r².h
Replacing the values
Idea 2: liquid volume
40% of 40 000
Idea 3: Transforming from cm³ to liters
For water, 1 000 cm³ = 1 liter
So 16 000 cm³ = 16 l
Conclusion
Therefore, 40% of the cylinder volume is equivalent to 16 liters.
Exercise 12
(FGV-SP) A straight circular cylinder of height equal to the base diameter is inscribed in a straight circular cone. The cone has a diameter of 10, height 12 and its axis of revolution coincides with that of the cylinder.
The diameter of the cylinder base is equal to
a) 16/3
b) 11/60
c) 6.
d) 4/25.
e) 7.
Correct answer: b) 60 / 11
Resolution
Idea 1: illustrating and identifying similar triangles
Making a southern section, or looking at a side view:
Taking a half of the figure, we have:
The larger triangle formed by the half cone is similar to the smaller green one because their angles are equal. (case A, A,A).
Idea 2: using proportions
On the left side we divide the height of the cone by half of its base, in this case, 5.
On the right side we divide the height of the green triangle, which is D, by its base, which is 5, minus half the diameter of the cylinder, which is D/2.
Adjustment on the right side, we have:
We can now multiply cross
5. 2D = 12. (20 - D)
10D = 120 - 12D
22D = 120
D = 120 / 22
D = 60 / 11
Conclusion
In this way, the diameter of the cylinder base is equal to 60/11.
Exercise 13
(PUC-PR). A drug that dilates the vessels and arteries in the human body is given and increases the diameter of a given artery by 20%.
Since the artery resembles a straight circular cylinder, blood flow in this artery increases by
a) 10%
b) 20%
c) 21%
d) 40%
e) 44%
Correct answer: e) 44%
Resolution
Flux is the amount of mass that passes through an area. In this case, the amount of blood that passes through the section of the artery.
Idea 1: The area of the artery section before the drug
Since the radius is half the diameter, we can write:
Idea 2: The area after the drug
To increase by 20%, we multiply D by 1.2.
Idea 3: comparing before and after areas
For this we will divide area 2 by 1
Eliminating similar terms
Conclusion
Thus, D2 was multiplied by 1.44, that is, an increase of 44%.
learn more about cylinders.
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