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The difference between outside and inside surfaces of a cylindrical metallic pipe 14 cm long is 44 cm2. If the pipe is made of 99 cm3 of metal, find the outer and inner radii of the pipe.

Answer
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Hint: First we will first assume that the outer and inner radii of the pipe are R cm and r cm respectively and his the height of the cylinder. We use the formula of the surface area of the cylinder is 2πrh, where r is the radius of the circle and h is the height of the cylinder to find the difference of the outside and inside surface area of a cylindrical pipe using the outer and inner radii. Then we will use the formula of the volume of the cylinder is πr2h, where r is the radius of the circle and h is the height of the cylinder to find the difference of the outside and inside volume of a cylindrical pipe using the outer and inner radii. Then substituting the values in the obtained equation to find the required values.

Complete step by step Answer:

We are given that the difference between outside and inside surfaces of a cylindrical metallic pipe 14 cm long is 44 cm2.

Let us assume that the outer and inner radii of the pipe are R cm and r cm respectively and his the height of the cylinder.

We know that the formula to find the surface area of the cylinder is 2πrh, where r is the radius of the circle and h is the height of the cylinder

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Finding the difference of the outside and inside surface area of a cylindrical pipe using the outer and inner radii, we get
2πR(14)2πr(14)=4428πR28πr=4428π(Rr)=44

Substituting the value of π in the above equation, we get
28×227×(Rr)=4428×22(Rr)=4488(Rr)=44

Dividing the above equation by 88 on both sides, we get
88(Rr)88=4488Rr=12 .......eq.(1)

We know that the formula to find the volume of the cylinder is πr2h, where r is the radius of the circle and h is the height of the cylinder

Since we are given that the pipe of metal is 99 cm cube, which is the volume of the pipe, by finding the difference of the outside and inside volume of a cylindrical pipe using the outer and inner radii, we get
πR2(14)πr2(14)=9914π(R2r2)=99

Substituting the value of π in the above equation, we get
14×227(R2r2)=9944(R2r2)=99

Dividing the above equation by 44 on both sides, we get
44(R2r2)44=9944R2r2=94(Rr)(R+r)=94

Using equation (1) in the above equation, we get
R+r2=94

Multiplying the above equation by 2 on both sides, we get
R+r=92 ......eq.(2)

Adding the equation (1) and equation (2), we get
Rr+R+r=12+922R=1022R=5

Dividing the above equation by 2 on both sides, we get
2R2=52R=52 cm

Substituting the value of R in the equation (2), we get
52+r=92r=9252r=42r=2 cm

Thus, the outer radius is 52 cm and the inner radius is 2 cm.

Note: In solving this type of question, the key concept is to assume the inner and outer radii by some distinct variables and also remember the formulae of the circumference of the circle, area of the cylinder, and the volume of the cylinder. After that, the question is really simple if followed properly.