Question

the base and height of a cylinder and cone are same. the ratio of their CSA is 8:5. Prove that the ratio of the radius and height be 3:4
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Answer 1

Step-by-step explanation:

Given:

• Base and height of a cylinder and cone are same.
• Ratio of their curved surface area is 8 : 5.

To Prove:

• Ratio of their radius and height is 3 : 4.

Solution: Let the base radius and height of cylinder be r and h respectively. Therefore,

➟ Radius of cone = r

➟ Height of cone = h

➟ Slant height of cone = l = √r² + h²

As we know that

★ CSA of Cylinder = 2πrh ★

★ CSA of Cone = πrl ★

Let's take it as

• CSA¹ = 2πrh
• CSA² = πrl

A/q

• Ratio is 8 : 5.

$\implies{\rm }$ 2πrh/πrl = 8/5

$\implies{\rm }$ 2πrh/πr√r² + h² = 8/5

$\implies{\rm }$ 2h/√r² + h² = 8/5

• {squaring both sides}

$\implies{\rm }$ (2h/√r² + h²)² = (8/5)²

$\implies{\rm }$ 4h²/r² + h² = 64/25

$\implies{\rm }$ h²/r² + h² = 16/25

By cross multiplication

➼ 25h² = 16r² + 16h²

➼ 25h² – 16h² = 16r²

➼ 9h² = 16r²

➼ 3h = 4r

➼ 3 = 4r/h

➼ 3/4 = r/h

➼ 3 : 4 = r : 4

$\large\bold{\texttt {Proved }}$

Answer 2

♧Answer♧

Consider the curved surface area of cylinder and cone as 8x and 5x

So we get

2π rh = 8x....(1)

πr_/(h²+r²) = 5x.....(2)

By squaring equation (1)

(2πrh)² = (8x)²

So we get 

4π²r²h² = 64x²....(3)

By squaring equation (2)

π²r²(h2+r2)=25x².....(4)

Dividing equation (3) by (4)

π²r²(h²+r²)/4π²r²h2² =25x²/64x²

On further calculation

(h2+r2)/h2=16/25

It can be written as

9h²=16r²

So we get,

r²/h² = 9/16

By taking square root

r/h = 3/4

r:h = 3:4

Therefore, it is proved that the radius and height of each has the ration 3:4.