Question

${\boxed {\LARGE{\mathcal {Question:-}}}}$
The TSA of a right circular cylinder is $\bf 701\dfrac {1}{4}cm^2.$.its LSA is $\bf 528 cm^2$

Find its volume ​

Answer 1

Given, A right circular cylinder, whose

$\tt{TSA = \bf 701\dfrac {1}{4} \: \: cm^2} \\ \\ \tt LSA = \: \: \bf 528 \: cm^2$

We know, the formulas below ↓

$\text{TSA of cylinder} \tt=2\pi r(r + h) \\ \\ \text{CSA of cylinder} \tt = 2\pi rh$

Here,

$\implies\bf701 \frac{1}{4} = 2\pi r(r + h) \\ \\ \implies\bf \frac{2805}{4} = 2\pi r(r + h) \\ \\ \implies\bf \frac{2805}{4} = {2\pi r}^{2} + 2\pi rh \\ \\ \implies\bf \frac{2805}{4} = 2\pi {r}^{2} + 528 \\ \\ \implies\bf \frac{2805 }{4} - 528 = 2\pi {r}^{2} \\ \\ \implies\bf \frac{2805 - 2112}{8} = \pi {r}^{2} \\ \\ \implies\bf \frac{693}{8} = \pi {r}^{2} \\ \\ \implies\bf \pi {r}^{2} = \frac{693}{8} \: {cm}^{2}$

So,

$\implies\bf {r}^{2} = \frac{693 \times 7}{8 \times 22} \\ \\ \implies\bf {r}^{2} = 27.5625 \\ \\ \implies\bf r = \sqrt{27.5625} \\ \implies\bf r = 5.25 \: cm$

Now,

$\implies\bf \frac{2805}{4} = 2\pi {r}^{2} + 2\pi rh \\ \\ \implies\bf \frac{2805}{4} = 2 \bigg( \frac{693}{8} \bigg) + 2\bigg( \frac{22}{7} \times 5.25 \bigg) \times h \\ \\\implies\bf h = \frac{ \frac{2805}{4} - \frac{693}{4} }{2\big( \frac{22}{7} \times 5.25 \big) } \\ \\ \implies\bf h = \frac{ \frac{2112}{4} }{2(16.5)} \\ \\ \implies\bf h = \frac{2112}{8(16.5)} \\ \\ \implies\bf h = \frac{2112}{132} = 16 \: \: cm$

Hence,

Volume of cylinder = π r²h

$\implies\bf \frac{22}{7} \times {(5.25)}^{2} \times 16 \\ \\ \implies\bf\frac{22}{7} \times 5.25 \times 5.25 \times 16 \\ \\ \implies\bf 22 \times0.75 \times 5.25 \times 16 \\ \implies\bf138 6 \: \: {cm}^{3}$