The quantity of heat required to raise the temperature of a body is given by $\Delta Q = mc\Delta T$. The mass of a solid sphere is $m = \text{Volume} \times \text{Density} = \frac{4}{3}\pi r^3 \rho$. Therefore, $\Delta Q = \frac{4}{3}\pi r^3 \rho c \Delta T$. For both copper spheres, the density $\rho$, specific heat capacity $c$, and the change in temperature $\Delta T$ ($1\text{ K}$) are the same. Thus, the heat required is directly proportional to the cube of the radius: $\Delta Q \propto r^3$. The ratio of the quantities of heat required is: $\frac{\Delta Q_1}{\Delta Q_2} = \left(\frac{r_1}{r_2}\right)^3$ Given $r_1 = 1.5 r_2 = \frac{3}{2} r_2$, we have $\frac{r_1}{r_2} = \frac{3}{2}$. Substituting this value: $\frac{\Delta Q_1}{\Delta Q_2} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$