A solid is in the form of a right circular cylinder with hemispherical ends. The total length of the solid is 35 cm. The diameter of the cylinder is one-fourth of its height. The surface area of the solid is :

1. 462 cm²

2. 693 cm²

3. 750 cm²

4. 770 cm²

From figure,

Height of cylinder be h cm

Given, Diameter of cylinder = $\dfrac{1}{4}$ h

Radius of cylinder = Radius of hemisphere = r = $\dfrac{\text{diameter}}{2} = \dfrac{\dfrac{1}{4} \times \text{h}}{2} = \dfrac{\text{h}}{8}$

Height of cylinder, h = Total height - (2 × Radius of hemisphere)

$\Rightarrow \text{h} = 35 - 2 \times \dfrac{\text{h}}{8} \\[1em] \Rightarrow \text{h} = 35 - \dfrac{\text{h}}{4} \\[1em] \Rightarrow \text{h} + \dfrac{\text{h}}{4} = 35 \\[1em] \Rightarrow \dfrac{4 \text{h} + \text{h}}{4} = 35 \\[1em] \Rightarrow \dfrac{5 \text{h}}{4} = 35 \\[1em] \Rightarrow 5\text{h} = 35 \times 4 \\[1em] \Rightarrow 5\text{h} = 140 \\[1em] \Rightarrow \text{h} = \dfrac{140}{5} \\[1em] \Rightarrow \text{h} = 28 \text{ cm.}$

∴ r = $\dfrac{\text{h}}{8} = \dfrac{28}{8} = 3.5 \text{ cm}$

Surface area of solid = 2 × 2πr² + 2πrh

= πr(4r + 2h)

$= \dfrac{22}{7} \times 3.5 (4 \times 3.5 + 2 \times 28) \\[1em] = 22 \times 0.5 (14 + 56) \\[1em] = 11 \times 70 \\[1em] = 770 \text{ cm}^2.$

Hence, option 4 is the correct option.