Question

The given figure shows the cross-section of a cone, a cylinder and a hemisphere all with the same diameter 10 cm, and the other dimensions are as shown.

Then, select the statements that are true.

• A. Total volume of the solid = 1519.0476 $c{m}^3$
• B. Total surface area of the solid = 750 $c{m}^2$
• C. Total surface area of the solid = 238.57 $c{m}^2$
• D. Total volume of the solid = 1919.0476 $c{m}^3$

Answer 1

The correct option is A

Total volume of the solid = 1519.0476 $c{m}^3$

Diameter = 10 cm

Therefore, radius (r) = 5 cm

Height of the cone (h) = 12 cm

Height of the cylinder = 12 cm

l = $\sqrt{h^2+r^2}$ = $\sqrt{{12}^2+5^2}$ = $\sqrt{144+25}$ = $\sqrt{169}$ = 13 cm

(i) Total surface area of the solid

= $\pi$rl + 2$\pi$rh + 2$\pi r^2$

= $\pi$(l + 2h + 2r)

= $\frac{22}{7}\times$5[13 + (2$\times$12) + 2$\times$5]

= $\frac{110}{7}$[13+24+10]

= $\frac{110}{7}\times 47$

= $\frac{5170}{7}$

= 738.57 $c{m}^2$

(ii) Total volume of the solid

= $\frac{1}{3}\pi r^{2}h$ + $\pi r^{2}h$ + $\frac{2}{3}\pi r^3$

= $\pi r^2[\frac{1}{3}h+h+\frac{2}{3}r]$

= $\frac{22}{7}\times 5\times 5[\frac{1}{3}\times 12+12+\frac{2}{3}\times 5]$

= $\frac{550}{7}[4+12+\frac{10}{3}]$

= $\frac{550}{7}\times \frac{58}{3}$

= $\frac{31900}{21}$

= 1519.0476 $c{m}^3$