P and Q are two cylinders having equal total surface areas The radius of each cylinder is equal to - Maths - Surface Areas and Volumes

Question

P and Q are two cylinders having equal total surface areas The
radius of each cylinder is equal to the hieght of the other The sum
of the volumes of both the cylinders is 250π
cm3 Find the sum of their curved
surface areas (in cm2)

Ankush Jain · Accepted Answer

For cylinder P: Let its height be h, cm and radius be r, cm For cylinder Q: Let the height be h2 cm and radius be r2 cm Given,Total surface area of cylinder P = Total surface area of cylinder Q $\Rightarrow 2 π r_{1} {(h_{1} + r_{1}) = 2 π r_{2} (h_{2} + r_{2}) \Rightarrow r_{1} h_{1} + r_{1}^{2} = r_{2} h_{2} + r_{2}^{2}}_{} \Rightarrow r_{1}^{2} - r_{2}^{2} = r_{2} h_{2} - r_{1} h_{1} h_{2}^{2} - h_{1}^{2} = h_{1} h_{2} - h_{2} h_{1} [Given r_{1} = h_{2} and r_{2} = h_{1}] \Rightarrow h_{2}^{2} = h_{1}^{2} or h_{2} = h_{1} (1)$ Again, volume of P + volume of Q = 250Ï€ $\Rightarrow π r_{1} {}^{2}{h_{1} + π r_{2}^{2}} h_{2} = 250 π \Rightarrow π h_{2} {}^{2}{h_{1} + π h_{1}^{2}} h_{2} = 250 π [As r_{1} = h_{2} and r_{2} = h_{1}] \Rightarrow π (h_{2}^{3} + h_{2}^{3}) = 250 π [using (1)] \Rightarrow 2 h_{2}^{3} = 250 h_{2}^{3} = 125 or h_{2} = \sqrt[3]{125} cm = 5 cm$ Now, h1 = h2 = 5 cm and r1 = r2 = 5 cm [âˆµ r1 = h2 and r2 = h1] Now, curved surface area of P + curved surface area of Q $= 2 π r_{1} h_{1} + 2 π r_{2} h_{2} = 2 π [5 \times 5 + 5 \times 5] {cm}^{2} = 2 π [5 0] = 100 π {cm}^{2}$

P and Q are two cylinders having equal total surface areas. The radius of each cylinder is equal to the hieght of the other. The sum of the volumes of both the cylinders is 250π cm3 . Find the sum of their curved surface areas. (in cm2).

P and Q are two cylinders having equal total surface areas. The radius of each cylinder is equal to the hieght of the other. The sum of the volumes of both the cylinders is 250π cm³ . Find the sum of their curved surface areas. (in cm²).