Surface Areas and Volumes: 9 Math
NCERT Exemplar Exercise 13.4: 9th math
Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math Question (1) A cylindrical tube opened at both the ends is made of iron sheet which is 2 cm thick. If the outer diameter is 16 cm and its length is 100 cm, find how many cubic centimeter of iron has been used in making the tube?
Solution
Given, length of the cylindrical tube = 100 cm
Thickness of tube = 2 cm
Outer diameter of the tube = 16 cm
Tube is opened at both the ends, then cubic centimetre of iron used in making of tube = ?
Since, the outer diameter of the tube = 16 cm and thickness of the tube = 2 cm
Thus, outer radius of the tube = 16/2 = 8 cm
And, therefore, inner radius of the tube = outer radius – thickness
= 8 cm – 2 cm = 6 cm
And, here length of the tube = height of the tube = 100 cm
Now, we know that, Volume of a cylinder = ℼ r2 h
Thus, outer volume of the given cylindrical pipe
= ℼ (8 cm)2 × 100 cm
= ℼ × 64 cm2 × 100 cm
= ℼ × 6400 cm3
Thus, total volume of the tube = ℼ × 6400 cm3
Now, the inner volume of the given cylindrical tube
= ℼ × (6 cm)2 × 100 cm
= ℼ 36 cm2 × 100 cm
= ℼ 3600 cm3
Thus, inner volume of the tube = ℼ 3600 cm3
Now, iron used to make the tube = outer volume of the tube – inner volume of the tube
= ℼ × 6400 cm3 – ℼ 3600 cm3
= ℼ (6400 cm3 – 3600 cm3)
= `22/7` × 2800 cm3
= 22 × 400 cm3
= 8800 cm3
Thus, iron used in making of given cylindrical tube
= 8800 cm3 Answer
Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math Question (2) A semi circular sheet of metal of diameter 28 cm is bent to form an open conical cup. Find the capacity of the cup.
Given, diameter of the semi circular metal sheet = 28 cm
Thus, radius of the semi circular (hemispherical) metal sheet = 28/2 = 14 cm
And as per question, the given semi-circular metal sheet is bent to form an open conical cup.
Thus, height of the conical cup = radius of the metal sheet = 14 cm
And, radius of the conical cup = 14 cm
Thus, capacity, i.e. volume of the conical cup formed using semi-circular metal sheet = ?
We know that, volume of a cone = 1/3 ℼ r2 h
Thus, volume of the conical cup formed after bending of given semi-circular metal sheet
= `1/3xx22/7` × ( 14 cm )2 × 14 cm
= `(1xx22)/3` × 14 cm × 14 cm × 2 cm
= 22/3 × 392 cm3
= 2874.66 cm3
Thus, capacity of the conical cup formed after bending of given semi-circular metal sheet
= 2874.66 cm3 Answer
Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math Question (3) A cloth having an area of 165 m2 is shaped into the form of a conical tent of radius 5m.
(i) How many students can sit in the tent if a student, on an average, occupies 5/7 m2 on the ground?
(ii) Find the volume of the cone.
Solution
Given, area of cloth = 165 m2
And, this cloth is shaped into a conical tent.
The radius of conical tent = 5 m
And, each student occupies 5/7 m2 of space in the tent
Thus, total number of students who can sit in the tent = ?
And volume of the conical tent = ?
(i) How many students can sit in the tent if a student, on an average, occupies 5/7 m2 on the ground?
And, radius of the conical tent = 5 m
Here it is clear that students will sit at the floor of the conical tent, and the base of the tent is circular.
Now, we know that, area of a circle = ℼ r2
Thus, area of the base of the given conical tent = ℼ (5 m)2
= ℼ 25 m2
Now, since in 5/7 m2 number of student can sit = 1
Thus, in 1 m2, the number of students can sit = 1 / 5/7 = 7/5
Thus, in ℼ 25 m2, the number of students can sit
`=7/5xx22/7xx25`
= 22 × 5 = 110
Thus, total number of students can sit in the given tent = 110 Answer
(ii) Find the volume of the cone.
Here, area of the cloth from which tent is made = 165 m2 = curved surface area of the tent
And, radius of the tent = 5 m
Calculation of slant height of the cone
Now, let the slant height of the tent `=l`
Now, we know that, Curved surface area of a cone `=pi\ r\ l`
Thus, curved surface area of the given tent
= 165 m2 = 22/7 × 5 m × `l`
`=>l = (165 m^2xx7)/(22xx5)`
⇒ slant height (`l`) of the tent = 10.5 m
Calculation of height of the conical tent
Now, we know that, (slant height)2 = r2 + h2
⇒ (10.5 m)2 = (5 m)2 + h2
⇒ 110.25 m = 25 m2 + h2
⇒ h2 = 110.25 m2 – 25 m2
⇒ h2 = 85.25 m2
⇒ h = 9.233 m
Thus, height of the conical tent = 9.233 m
Calculation of volume of the given conical tent
Now, we know that, Volume of a Cone = 1/3 ℼ r2 h
Thus, volume of the given conical tent
= 1/3 × 22/7 × (5 m)2 × 9.233 m
= 22/21 × 25 m2 × 9.233 m
= 22/21 × 230.825 m3
= 241.816 m3
Thus, volume of the given tent = 241.816 m3 Answer
Solution of NCERT Exemplar Exercise 13.4 Surface Areas And Volumes Class 9 Math Question (4) The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Calculate the volume of water pumped into the tank.
Solution
Given, the diameter of the hemispherical tank = 14 m
Thus, radius of the hemispherical tank = 14/2 = 7 m
Water already in the tank = 50 kilolitre = 50 m3
Water is pumped to fill the tank to its capacity
Thus, volume of water pumped into the tank = ?
Here, volume of water pumped in the tank = volume of tank – water already in the tank
Now, we know that, Volume of a hemisphere = 2/3 ℼ r3
Thus, volume of the given hemispherical tank
= 2/3 × 22/7 × (7m)^3
= 2/3 × 22/7 × 7 m × 7 m × 7 m
= 2/3 × 22 × 1 m × 7 m × 7 m
= 44/3 × 49 m3
= 718.66 m3
Thus, capacity of the given hemispherical tank = 718.66 m3
Now, water pumped into the tank = capacity of tank – water already in the tank
= 718.66 m3 – 50 m3
= 668.66 m3
Thus, water pumped into the given hemispherical tank = 668.66 kilolitre Answer
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