A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8 :5, the ratio of the radius and their height. एक बेलन और शंकु की आधार की त्रिज्या तथा इनकी ऊंचाई समान हैं | यदि इनके वक्रपृष्ठ के क्षेत्रफल का अनुपात 8 : 5 है तो इनकी त्रिज्या और ऊंचाई का अनुपात ज्ञात कीजिये ?

Let radius = r and height = h. Curved surface area of cylinder = 2πrh. Curved surface area of cone = πrl where l = √(r²+h²). Given 2πrh / πr√(r²+h²) = 8/5, we get 2h / √(r²+h²) = 8/5. Squaring: 4h² / (r²+h²) = 64/25, which simplifies to 100h² = 64r² + 64h², giving 36h² = 64r². Taking square root: 6h = 8r, so r/h = 6/8 = 3/4. The ratio of radius to height is 3:4.