# Mensuration Questions and Answers updated daily – Aptitude

Mensuration Questions: Solved 782 Mensuration Questions and answers section with explanation for various online exam preparation, various interviews, Aptitude Category online test. Category Questions section with detailed description, explanation will help you to master the topic.

## Mensuration Questions

1. Smallest side of a right angled triangle is 13 cm less than the side of a square of perimeter 72 cm. Second largest side of the right angled triangle is 2 cm less than the length of the rectangle of area 112 cmÂ² and breadth 8 cm. What is the largest side of the right angled triangle?

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Correct Ans:13 cm

Explanation:

Explanation :

Side of square = 72/4 = 18 cm

Smallest side of the right angled triangle = 18 â€“ 13 = 5 cm

Length of rectangle = 112/8 = 14 cm

Second side of the right angled triangle = 14 â€“ 2 = 12 cm

Hypotenuse of the right angled triangle = √(25 + 144) =

Side of square = 72/4 = 18 cm

Smallest side of the right angled triangle = 18 â€“ 13 = 5 cm

Length of rectangle = 112/8 = 14 cm

Second side of the right angled triangle = 14 â€“ 2 = 12 cm

Hypotenuse of the right angled triangle = √(25 + 144) =

**13cm**
Workspace

2. The length of a rectangle is 3/5th of the side of a square. The radius of a circle is equal to side of the square. The circumference of the circle is 132 cm. What is the area of the rectangle, if the breadth of the rectangle is 15 cm?

SHOW ANSWER

Correct Ans:189 cm

^{2}Explanation:

Explanation :

Circumference of the circle = 132

2Ï€R = 132;

R = 21 cm

Side of square = 21 cm

Length of the rectangle = 3/5 * 21 = 63/5

Area of the rectangle = 63/5 * 15 =

Circumference of the circle = 132

2Ï€R = 132;

R = 21 cm

Side of square = 21 cm

Length of the rectangle = 3/5 * 21 = 63/5

Area of the rectangle = 63/5 * 15 =

**189 cm**^{2}
Workspace

3. The perimeter of a square is equal to twice the perimeter of a rectangle of length 10 cm and breadth 4 cm. What is the circumference of a semi-circle whose diameter is equal to the side of the square?

SHOW ANSWER

Correct Ans:36 cm

Explanation:

Explanation :

Perimeter of square = 2(l + b)

= 2 * 2(10 + 4) = 2 * 28 = 56 cm

Side of square = 56/4 = 14 cm

Radius of semi circle = 14/2 = 7cm

Circumference of the semi-circle = 22/7 * 7 + 14 =

Perimeter of square = 2(l + b)

= 2 * 2(10 + 4) = 2 * 28 = 56 cm

Side of square = 56/4 = 14 cm

Radius of semi circle = 14/2 = 7cm

Circumference of the semi-circle = 22/7 * 7 + 14 =

**36 cm**
Workspace

4. The length of a rectangle is reduced by 30%. By what percent would the width have to be increased to maintain the original area?

SHOW ANSWER

Correct Ans:42.86%

Explanation:

Explanation :

----> Width = 30*(100/100 ) - 30

----> = (3000/70) =

----> Width = 30*(100/100 ) - 30

----> = (3000/70) =

**42.86%**
Workspace

5. Six spherical cannon balls are tightly packed into a rectangular box in one layer. Each row has two cannon balls and each column has three. What part of the box is empty?

SHOW ANSWER

Correct Ans:(10/21)

Explanation:

----> Let the diameter of each ball be 2r.

----> Length of the box = 3*2r = 6r

----> Breadth = 2*2r = 4r

----> Height = 2r

----> Volume = 6r * 4r * 2r = 48r

----> Volume of 6 balls = 6 * (4/3) * (22/7) * r

----> The area of empty space = 48 r

----> = (160 r

----> The required fraction = ((160 r

---->

----> Length of the box = 3*2r = 6r

----> Breadth = 2*2r = 4r

----> Height = 2r

----> Volume = 6r * 4r * 2r = 48r

^{3}----> Volume of 6 balls = 6 * (4/3) * (22/7) * r

^{3}= (176 r^{3}/7)----> The area of empty space = 48 r

^{3}- (176 r^{3}/7)----> = (160 r

^{3}/7)----> The required fraction = ((160 r

^{3}/7)/48 r^{3})---->

**= (10/21)**
Workspace

6. A circular wire of radius 49 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 4:7. The smaller side of the rectangle is ?

SHOW ANSWER

Correct Ans:56cm

Explanation:

Explanation :

-----> circumference = 2*(22/7)*49 = 308 cm

-----> length of rectangle sides are 4x, 7x.

-----> circumference = 2*(4x+7x)

-----> 308 = 22x

-----> X = (308/22) = 14

-----> smaller side of rectangle = 4x = 4*14 =

-----> circumference = 2*(22/7)*49 = 308 cm

-----> length of rectangle sides are 4x, 7x.

-----> circumference = 2*(4x+7x)

-----> 308 = 22x

-----> X = (308/22) = 14

-----> smaller side of rectangle = 4x = 4*14 =

**56 cm**
Workspace

7. In a swimming pool measuring 90 m x 40 m, 150 men take a dip. If the average displacement of water by a man is 8 m cube, What will be the rise in water level?

SHOW ANSWER

Correct Ans:(1/3) m

Explanation:

Given, Average displacement of water by a man = 8 m

Then, total volume of water displaced by 150 men = 150 * 8 m

= 1200 m

Total volume of the swimming pool (Volume of cuboid) = l * b * h

= 90 * 40 * h

= 3600 * h

---> (here

Now, Total volume of the swimming pool = total volume of water displaced by 150 men

---> 3600 * h = 1200

---> h = 12/36

--->

Thus,

^{3}Then, total volume of water displaced by 150 men = 150 * 8 m

^{3}= 1200 m

^{3}Total volume of the swimming pool (Volume of cuboid) = l * b * h

= 90 * 40 * h

= 3600 * h

---> (here

**h is the height by which the water level rises**)Now, Total volume of the swimming pool = total volume of water displaced by 150 men

---> 3600 * h = 1200

---> h = 12/36

--->

**h = 1/3 m**Thus,

**the water level rises by 1/3 m**
Workspace

8. If the slant height of a right pyramid with square base is 4 metre and the total slant surface of the pyramid is 12 square metre, then what is the ratio of total slant surface and area of the base?

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Correct Ans:(16 : 3)

Explanation:

Given, Base of the right pyramid is square

Slant height = h = 4 metre

Total slant surface area of the pyramid = 12 square metre

W.K.T:

---> 12 = (1/2) * 4a * 4

Where, a = side of the square base

---> 12 = 8a

---> a = 12/8

---> a = 3/2

Thus, the

Therefore,

= (3/2)

=

Now,

= 12/(9/4)

= (12 * 4)/9

= 16/3

=

Slant height = h = 4 metre

Total slant surface area of the pyramid = 12 square metre

W.K.T:

**Total slant surface area (Lateral surface area) of the square Pyramid = (1/2) * Perimeter of the square base * slant height**---> 12 = (1/2) * 4a * 4

Where, a = side of the square base

---> 12 = 8a

---> a = 12/8

---> a = 3/2

Thus, the

**side of the square base = a = 3/2 metre**Therefore,

**Area of the base**(i.e, area of square base) = a^{2}= (3/2)

^{2}=

**9/4**m^{2}Now,

**Required ratio**= Total slant surface area/Area of the base= 12/(9/4)

= (12 * 4)/9

= 16/3

=

**16 : 3**
Workspace

9. The base of a right prism is a quadrilateral ABCD, given that AB = 9 cm, BC = 14 cm, CD = 13 cm, DA = 12 cm and ∠DAB = 90°, If the volume of the prism be 2070 cm

^{3}, then what is the area of the lateral surface?SHOW ANSWER

Correct Ans:720 cm

^{3}Explanation:

In ∆ABD,

= (1/2) * 9 * 12

=

In ∆ABD, by Pythagoras theorem,

BD = √[AB

= √[9

= √[81 + 144]

= √[225]

= 15 cm

In ∆BCD,

Semi-perimeter, S = (BC + CD + DB)/2

= (14 + 13 + 15)/2

= 42/2

= 21 cm

Now,

Where, S = Semi-perimeter = 21cm

a = BC

b = CD

c = DB

Area of ∆BCD = √[21(21 - 14)(21 - 13)(21 - 15)]

= √[21(7)(8)(6)]

= √[7056]

=

Now,

= 54 + 84

=

= 2070/138

=

= 48 * 15

=

**Area of ∆ABD**= (1/2) * AB * AD= (1/2) * 9 * 12

=

**54 cm**^{2}In ∆ABD, by Pythagoras theorem,

BD = √[AB

^{2}+ AD^{2}]= √[9

^{2}+ 12^{2}]= √[81 + 144]

= √[225]

= 15 cm

In ∆BCD,

Semi-perimeter, S = (BC + CD + DB)/2

= (14 + 13 + 15)/2

= 42/2

= 21 cm

Now,

**Area of ∆BCD = √[S(S - a)(S - b)(S - c)]**Where, S = Semi-perimeter = 21cm

a = BC

b = CD

c = DB

Area of ∆BCD = √[21(21 - 14)(21 - 13)(21 - 15)]

= √[21(7)(8)(6)]

= √[7056]

=

**84 cm**^{2}Now,

**Area of quadrilateral ABCD = Area of base of Prism**= Area of ∆ABD + Area of ∆BCD= 54 + 84

=

**138 cm**^{2}**Height of Prism = Volume/Area of base**= 2070/138

=

**15 cm****Perimeter of base of the Prism**= sum of the bases = 9 + 14 + 13 + 12 = 48 cm**Area of lateral surface = Perimeter of the base * Height of the prism**= 48 * 15

=

**720 cm**^{2}
Workspace

10. 18 cylindrical water bottles with height same as that of the radius are emptied into a spherical earthen pot. The bottles fill half of the initially empty earthen pot. Find the ratio of the radius of the cylinder and that of the pot?

SHOW ANSWER

Correct Ans:(1:3)

Explanation:

18 cylindrical water bottles with height same as that of the radius are emptied into a spherical earthen pot. The bottles fill half of the initially empty earthen pot. Find the ratio of the radius of the cylinder and that of the pot:

Reference:

---> Let the height and radius of bottles be r

---> Volume of all bottles = 18(πr

---> Let the radius of pot be P.

---> Half of the volume of the pot = (2πP

---> (2πP

---> πP

---> P = 3r

---> Ratio = (1:3)

Reference:

---> Let the height and radius of bottles be r

---> Volume of all bottles = 18(πr

^{2}*r) = 18πr^{3}---> Let the radius of pot be P.

---> Half of the volume of the pot = (2πP

^{3}/3)---> (2πP

^{3}/3) = 18πr^{3}---> πP

^{3}= 27πr^{3}---> P = 3r

---> Ratio = (1:3)

**Hence the answer is : (1:3)**
Workspace

11. The radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3). Find the ratio of their volumes?

SHOW ANSWER

Correct Ans:( 32 : 25)

Explanation:

The radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3). Find the ratio of their volumes:

Reference:

---> Volume of a cylinder = πr

---> Volume of a cone = (1/3)πr

---> Ratio of volume of cylinder to cone = (πr

---> Ratio of volume of cylinder to cone

---> = (3 * ratio of radius

---> Given, radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3).

---> Ratio of volumes = 3 * (4/5)

---> Ratio of volumes = ( 32 : 25)

Reference:

---> Volume of a cylinder = πr

^{2}h---> Volume of a cone = (1/3)πr

^{2}h---> Ratio of volume of cylinder to cone = (πr

^{2}h/ (1/3)πr1^{2}h1 )---> Ratio of volume of cylinder to cone

---> = (3 * ratio of radius

^{2}* ratio of height )---> Given, radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3).

---> Ratio of volumes = 3 * (4/5)

^{2}* (2/3)---> Ratio of volumes = ( 32 : 25)

**Hence the answer is : ( 32 : 25)**
Workspace

12. A cone and a cylinder are of the same height. Their radii of their base are in ratio of (2 : 1). The ratio of their volumes is

SHOW ANSWER

Correct Ans:(4/3)

Explanation:

A cone and a cylinder are of the same height. Their radii of their base are in ratio of (2 : 1). The ratio of their volumes is

Reference:

----> Let radius of cylinder = x and radius of cone = 2x

----> Height of each = h

----> Required ratio = (Volume of cone/Volume of cylinder)

----> = ( (1/3) π4 x

----> = (4/3)

Reference:

----> Let radius of cylinder = x and radius of cone = 2x

----> Height of each = h

----> Required ratio = (Volume of cone/Volume of cylinder)

----> = ( (1/3) π4 x

^{2}h/ π x^{2}h )----> = (4/3)

**Hence the answer is : (4/3)**
Workspace

13. The total surface area of a cylinder is 2/9 times the total surface area of a sphere. If the ratio of the curved surface area of the cylinder is 2 : 3 by the difference between the curved surface area of the cylinder and the total surface area of the top and bottom of the cylinder, find the volume of the sphere. It is given that the volume of cylinder is 3234 cubic.cm.

SHOW ANSWER

Correct Ans:38808 cubic.cm

Explanation:

Given:

(Curved surface area of â€‹â€‹the cylinder - Total surface area of â€‹â€‹the top and bottom of the cylinder)/Curved surface area of â€‹â€‹the cylinder = 2 : 3

WKT,Curved surface area of â€‹â€‹the cylinder = 2πrh

Total surface area of top and bottom of the cylinder =2πr

(2πrh - 2πr

2πr(r - h)/2πrh = 2/3

(h - r)/h = 2/3

1 - (r/h) = 2/3

r/h = 1/3

Let r be 'x cm' and h be '3x cm'.

Also given, volume of cylinder is 3234 cubic cm.

Wkt, Volume of cylinder =πr

πr

(22/7)*x

x

x

x = 7 cm

Hence, r = 7 cm; h = 21 cm

Let the radius of sphere be 'R cm'.

WKT,Total surface area of â€‹â€‹a cylinder=2πrh +2πr

Total surface area of â€‹â€‹a sphere = 4πR

Total surface area of â€‹â€‹a cylinder is 2/9 times the total surface area of â€‹â€‹a sphere.

2πrh +2πr

2*(22/7)*7*[21 + 7] = (2/9)*4*(22/7)*R

R

R = 21 cm.

Volume of sphere = (4/3)πR

= (4/3) x (22/7) x 21 x 21 x 21

= 38808 cubic.cm.

(Curved surface area of â€‹â€‹the cylinder - Total surface area of â€‹â€‹the top and bottom of the cylinder)/Curved surface area of â€‹â€‹the cylinder = 2 : 3

WKT,Curved surface area of â€‹â€‹the cylinder = 2πrh

Total surface area of top and bottom of the cylinder =2πr

^{2}(2πrh - 2πr

^{2})/2πrh = 2/32πr(r - h)/2πrh = 2/3

(h - r)/h = 2/3

1 - (r/h) = 2/3

r/h = 1/3

Let r be 'x cm' and h be '3x cm'.

Also given, volume of cylinder is 3234 cubic cm.

Wkt, Volume of cylinder =πr

^{2}hπr

^{2}h = 3234(22/7)*x

^{2}*3x = 3234x

^{3}= (3234*7)/(22*3)x

^{3}= 343x = 7 cm

Hence, r = 7 cm; h = 21 cm

Let the radius of sphere be 'R cm'.

WKT,Total surface area of â€‹â€‹a cylinder=2πrh +2πr

^{2}Total surface area of â€‹â€‹a sphere = 4πR

^{2}Total surface area of â€‹â€‹a cylinder is 2/9 times the total surface area of â€‹â€‹a sphere.

2πrh +2πr

^{2}= (2/9)4πR^{2}2*(22/7)*7*[21 + 7] = (2/9)*4*(22/7)*R

^{2}R

^{2}= 441R = 21 cm.

Volume of sphere = (4/3)πR

^{3}= (4/3) x (22/7) x 21 x 21 x 21

= 38808 cubic.cm.

Workspace

14. A room is of dimensions 9*8*6.5 in meters. It has one door of dimensions 2 m x 1.5 m and three windows each of dimensions 1.5 m x 1 m. Find the cost of white washing the walls but excluding the door and windows, at Rs 3.80 per square metre.

SHOW ANSWER

Correct Ans:Rs. 811.3

Explanation:

Given, Length of the room = 9 m

Breadth of the room = 8 m

Height of the room = 6.5 m

W.K.T:

Area of the 4 walls of room = 2 (l + b) * h

= 2 (9 + 8) * 6.5

= 34 * 6.5

= 221 m

Area of 1 door = 2 * 1.5 = 3 m

Area of 1 window = 1.5 Ã— 1 = 1.5 m

Area of 3 windows = 3 Ã— 1.5 = 4.5 m

= 221 - 3 - 4.5

=

=

Breadth of the room = 8 m

Height of the room = 6.5 m

W.K.T:

**Lateral surface area of a Cuboid = 2 (l + b) * h**Area of the 4 walls of room = 2 (l + b) * h

= 2 (9 + 8) * 6.5

= 34 * 6.5

= 221 m

^{2}Area of 1 door = 2 * 1.5 = 3 m

^{2}Area of 1 window = 1.5 Ã— 1 = 1.5 m

^{2}Area of 3 windows = 3 Ã— 1.5 = 4.5 m

^{2}**Area required to white wash the wall**= Area of 4 walls - Area of 1door - Area of 3 windows= 221 - 3 - 4.5

=

**213.5 m**^{2}**Cost of white washing the walls**= 213.5 * 3.8=

**Rs. 811.3**
Workspace

15. The sum of circumference of a circle and perimeter of a rectangle is 480 cm. If the area of the rectangle is 168 cm

^{2}and ratio of radius of the circle and breadth of the rectangle is 19 : 2. Find the area of another circle whose radius is 28% of the diagonal of rectangle. (Note: Length and Breadth of rectangle must be integer)SHOW ANSWER

Correct Ans:154 cm

^{2}Explanation:

Given:

Area of the rectangle = 168 cm

Ratio of radius of the circle and breadth of the rectangle = 19 : 2

Let the radius of the circle be 19x and breadth of the rectangle be 2x.

WKT, Area of rectangle = l*b

l*b = 168

l*2x = 168

l = 84/x cm

WKT, circumference of a circle = 2π r

Perimeter of rectangle = 2(l + b)

Sum of circumference of a circle and perimeter of a rectangle = 480 cm

2π r + 2(l + b) = 480

2*(22/7)*19x + 2[(84/x) + 2x] = 480

36x

x = 7/2

So, breadth of rectangle = 2(7/2) = 7 cm

length of rectangle = 84/(7/2) = 24 cm

Diagonal of rectangle = √[24

= √[576 + 49]

= √625

= 25 cm

Let the radius of new circle be 'R'.

Radius of new circle is 28% of the diagonal of rectangle = (28/100)*25

= 7 cm.

Area of new circle = π R

= (22/7)*(7)

= 154 cm

Area of the rectangle = 168 cm

^{2}Ratio of radius of the circle and breadth of the rectangle = 19 : 2

Let the radius of the circle be 19x and breadth of the rectangle be 2x.

WKT, Area of rectangle = l*b

l*b = 168

l*2x = 168

l = 84/x cm

WKT, circumference of a circle = 2π r

Perimeter of rectangle = 2(l + b)

Sum of circumference of a circle and perimeter of a rectangle = 480 cm

2π r + 2(l + b) = 480

2*(22/7)*19x + 2[(84/x) + 2x] = 480

36x

^{2}- 140x + 49 = 0x = 7/2

So, breadth of rectangle = 2(7/2) = 7 cm

length of rectangle = 84/(7/2) = 24 cm

Diagonal of rectangle = √[24

^{2}+ 7^{2}]= √[576 + 49]

= √625

= 25 cm

Let the radius of new circle be 'R'.

Radius of new circle is 28% of the diagonal of rectangle = (28/100)*25

= 7 cm.

Area of new circle = π R

^{2}= (22/7)*(7)

^{2}= 154 cm

^{2}.
Workspace

16. If the sum of the length, breadth and height of a rectangular parallelepiped is 24 cm and the length of its diagonal is 15 cm, then its total surface area is

SHOW ANSWER

Correct Ans:351 cm

^{2 }Explanation:

If the sum of the length, breadth and height of a rectangular parallelepiped is 24 cm and the length of its diagonal is 15 cm, then its total surface area is

Reference :

----> l + b +h = 24

----> l

----> l

----> (l+b+h)

----> 24

----> 2(lb+bh+hl)=576 â€“ 225 = 351 cm

Reference :

----> l + b +h = 24

----> l

^{2}+ b^{2}+ h^{2}= 15^{2}----> l

^{2}+ b^{2}+ h^{2}= 225----> (l+b+h)

^{2}= l^{2}+ b^{2}+ h^{2}+ 2(lb+bh+hl)----> 24

^{2}= 225 + 2(lb+bh+hl)----> 2(lb+bh+hl)=576 â€“ 225 = 351 cm

^{2}**Hence the answer is : 351 cm**^{2}
Workspace

17. Two steel sheets each of length a1 and breadth a2 are used to prepare the surfaces of two right circular cylinders "“ one having volume v1 and height a2 and other having volume v2 and height a1. Then,

SHOW ANSWER

Correct Ans:a

_{2}v_{1}= a_{1}v_{2}Explanation:

Area of sheets = a

Area of sheets = Area of cylinder

-----> a

-----> For first cylinder, height = a

-----> Therefore,

-----> a

-----> r

-----> Similarly, for the second cylinder,

-----> r

-----> The ratio of volumes , ((π r

-----> (v

-----> (v

_{1}a_{2}Area of sheets = Area of cylinder

-----> a

_{1}a_{2}= 2πrh-----> For first cylinder, height = a

_{2}-----> Therefore,

-----> a

_{1}a_{2}= 2πr_{1}a_{2}-----> r

_{1}= (a_{1}/2π)-----> Similarly, for the second cylinder,

-----> r

_{2}= (a_{2}/2π)-----> The ratio of volumes , ((π r

_{1}^{2}a_{2})/(π r_{2}^{2}a_{1})) = (v_{1}/v_{2})-----> (v

_{1}/v_{2}) = (a_{1}/a_{2})-----> (v

_{1}/a_{1}) = (v_{2}/a_{2})**Hence the answer is : a**_{2}v_{1}= a_{1}v_{2}
Workspace

18. Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes?

SHOW ANSWER

Correct Ans:3:1

Explanation:

Ratio of volume = (1/3πr

We know that,

Height = 1 : 3

Radius = 3 : 1

Ratio of volume = 9/1 * 1/3 = 3/1 = 3 : 1

_{1}^{2}h_{1}) / (1/3πr_{2}^{2}h_{2})We know that,

Height = 1 : 3

Radius = 3 : 1

Ratio of volume = 9/1 * 1/3 = 3/1 = 3 : 1

Workspace

19. A sector of 120 degrees, cut out from a circle, has an area of 66/7 sq cm. Find the radius of the circle?

SHOW ANSWER

Correct Ans:3 cm

Explanation:

Ï€r

22/7 * r

r

r = 3 cm

^{2}* 120Â°/360Â° = 66/722/7 * r

^{2}* 1/3 = 66/7r

^{2}= 3*3r = 3 cm

Workspace

20. A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made. The total number of possible small lead balls is

SHOW ANSWER

Correct Ans:8000

Explanation:

Formula:

Volume of sphere = (4/3)Ï€r

1 cm = 10 mm

A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made.

Radius of small lead balls = 3/10 = 0.3 cm

Let n be the number of small lead balls

Therefore,

(4/3)Ï€(6)

(6*6*6*1000)/(3*3*3) = n

n = 8000

Volume of sphere = (4/3)Ï€r

^{3}1 cm = 10 mm

A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made.

Radius of small lead balls = 3/10 = 0.3 cm

Let n be the number of small lead balls

Therefore,

(4/3)Ï€(6)

^{3}= n * (4/3)Ï€(0.3)^{3}(6*6*6*1000)/(3*3*3) = n

n = 8000

Workspace

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