Question : If the radius of a hemispherical balloon increases from 4 cm to 7 cm as air is pumped into it, find the ratio of the surface area of the new balloon to its original.
Option 1: 16 : 21
Option 2: 49 : 16
Option 3: 20 : 49
Option 4: 21 : 12
Correct Answer: 49 : 16
Solution :
The surface area of a hemisphere = $3 \pi r^2$
The radius of the balloon before pumping air, $r_1$ = 4 cm
The radius of the balloon after pumping air, $r_2$ = 7 cm
The surface area of the balloon before pumping air, $SA_1$ = $3 \pi r_1^2$
The surface area of the balloon after pumping air, $SA_2$ = $3 \pi r_2^2$
The ratio of the surface areas of the balloon,
= $\frac{SA_2}{SA_1}$
= $\frac{3 \pi r_2^2}{3 \pi r_1^2}$
= $\frac{(r_2)^2}{(r_1)^2}$
= $(\frac{7}{4})^2$
= $\frac{49}{16}$
The ratio of the surface areas of the balloons = 49 : 16
Hence, the correct answer is 49 : 16.
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