Q) A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surface areas are in the ratio 8: 5, then find the ratio between the radius of their bases to their height.

[Q 28 – 30/1/3 – CBSE 2026 Question Paper]

Ans:

Step 1: Let’s draw a diagram for our better understanding of the question:

Here, Cylinder and the Cone with equal bases and equal heights.

Step 2: Let’s consider Radius of base = R and Height = H.

We know that the CSA of Cylinder, CSA_Cyl = 2 π RH

and CSA of Cone, CSA_Cone = π R L

and ∵ L = √ (R² + H²)

∴ CSA_Cone = π R √ (R² + H²)

Step 3: Given that the Curved surface areas of Cylinder and Cone are in the ratio 8 : 5

∴  CSA_Cyl : CSA_Cone  = 8 : 5

∴ 5 CSA_Cyl =  8 CSA_Cone  

Now, by substituting values from step 2, we get:

∴ 5 (2 π RH) =  8 (π R √ (R² + H²))

∴ 5 H =  4 √ (R² + H²)

By squaring on both sides, we get:

∴ (5 H)² =  (4 √ (R² + H²))²

∴ 25 H² =  16  (R² + H²)

∴ 25 H² – 16 H² =  16 R²

∴ 9 H² =  16 R²

∴ 3 H = 4 R

∴ R : H = 3 : 4

Therefore, the ratio between the radius of the base and the height is 3 : 4.

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