Question

A solid cylinder has total surface area of 462 sq cm.Its curved area is one third of its total surface area .Find the volume of the cylinder.Take π as 22÷7

Answer 1

TSA = 462 square cm

$2\pi \: r(h + r) = 462 \\ 2\pi \: rh + 2\pi {r}^{2} = 462 \\ 154 + 2\pi {r}^{2} = 462 \\ 2\pi {r}^{2} = 308 \\ {r}^{2} = 308 \times 7 \div (2 \times 22) \\ {r}^{2} = 49 \\ r = 7cm$

Also

CSA = 1/3 TSA

$2\pi \: rh = 154 \: \\ 2 \times \frac{22}{7} \times 7 \times h = 154 \\ h = 154 \div (2 \times 22) \\ h = 7 \div 2 = 3.5cm$

Volume of cylinder
$= \pi {r}^{2} h \\ = \frac{22}{7} \times 7 \times 7 \times 3.5 \\ = 7 \times 77 \\ = \: 539 \: cubic \: cm$

Answer 2

$\text{Given}~: \\ \text{CSA of the cylinder} = \frac{1}{3}(\text{TSA of the cylinder}) \\\implies \text{CSA of cylinder} = \frac{462}{3} = 154 \\ \\ \text{CSA is one third of TSA} \\\implies \frac{TSA}{CSA} = \frac{3}{1} \\\implies \frac{2\pi r(r+h)}{2\pi rh} = \frac{3}{1} \\\implies \frac{\cancel{2\pi r}(r+h)}{\cancel{2\pi r}h} = \frac{3}{1} \\\implies \frac{r+h}{h} = 3 \\\implies r+h = 3h \\\implies r = 2h \\ \\\text{putting value of r from above equation in formula of CSA} : \\\implies 2\pi 2h \times h = 154 \\\implies 4\pi h^2 = 154 \\\implies h^2 = \frac{154\times 7}{4 \times 22} \\\implies \sqrt{h^2} = \sqrt{\frac{7\times7}{4}} \\\implies h = \frac{7}{2} \\\implies r = \frac{7}{2} \times 2 = 7 \\\text{formula for volume of cylinder} = \pi r^2 h \\ \implies ~Volume ~=\frac{22 \times 7 \times 7 \times 7}{7\times 2} \\\implies \text{Volume of cylinder} = 539 cm^3$