A student measures the mass and radius of a solid sphere to...

The density of a sphere is given by the formula:

$\rho = \dfrac{M}{V} = \dfrac{M}{\dfrac{4}{3}\pi R^3}$

The maximum percentage error in density is given by the sum of the percentage errors of the individual quantities multiplied by their respective powers. Constants like $\dfrac{4}{3}\pi$ do not contribute to the error.

$\dfrac{\Delta \rho}{\rho} \times 100 = \left(\dfrac{\Delta M}{M} \times 100\right) + 3\left(\dfrac{\Delta R}{R} \times 100\right)$

Given that the percentage error in mass is $2\%$ and in radius is $1.5\%$:

$\dfrac{\Delta \rho}{\rho} \times 100 = 2\% + 3(1.5\%)$

$\dfrac{\Delta \rho}{\rho} \times 100 = 2\% + 4.5\% = 6.5\%$

Thus, the maximum percentage error in the calculated density is $6.5\%$.

Answer: $6.5\%$