AP TET Paper 2 Math and Science Model Paper 2

Concept:The entire volume of iron is conserved when the large ball is melted into 64 small balls.Surface area depends on the square of the radius, so the ratio is found by comparing $4\pi R^2$ with $64 \times 4\pi r^2$.Explanation:Let the radius of the larger ball be $R$ and the radius of each smaller ball be $r$.Volume of the larger ball: $V_L = \frac{4}{3}\pi R^3$Total volume of 64 smaller balls: $V_s = 64 \times \frac{4}{3}\pi r^3$Since the entire volume is used, $V_L = V_s$.$\frac{4}{3}\pi R^3 = 64 \times \frac{4}{3}\pi r^3$$R^3 = 64r^3$$R = 4r$Surface area of the larger ball: $A_L = 4\pi R^2 = 4\pi(4r)^2 = 64\pi r^2$Sum of surface areas of all smaller balls: $A_s = 64 \times 4\pi r^2 = 256\pi r^2$Required ratio: $\frac{A_L}{A_s} = \frac{64\pi r^2}{256\pi r^2} = \frac{1}{4} = 0.25$Answer:The required ratio is $\frac{1}{4}$, which is $0.25$.Hence, the correct option is A.