To solve this problem, we need to determine the ratio of the heat required to raise the temperature of two copper spheres.$\\$The heat required to raise the temperature of a solid sphere is given by:$\\ Q = mc\Delta T \\$where $m$ is the mass, $c$ is the specific heat capacity, and $\Delta T$ is the change in temperature.$\\$For a sphere, the mass $m$ can be expressed as:$\\ m = \rho V = \rho \left(\frac{4}{3}\pi r^3\right) \\$where $\rho$ is the density and $V$ is the volume of the sphere.$\\$Thus, the heat required for each sphere is:$\\ Q_1 = \rho \left(\frac{4}{3}\pi r_1^3\right) c \Delta T \\ Q_2 = \rho \left(\frac{4}{3}\pi r_2^3\right) c \Delta T \\$Given that $r_1 = 1.5 r_2,$ we substitute this into the expressions for $Q_1$ and $Q_2: \\ Q_1 = \rho \left(\frac{4}{3}\pi (1.5 r_2)^3\right) c \Delta T \\ Q_2 = \rho \left(\frac{4}{3}\pi r_2^3\right) c \Delta T \\$Simplifying $Q_1: \\ Q_1 = \rho \left(\frac{4}{3}\pi (3.375 r_2^3)\right) c \Delta T \\ Q_1 = 3.375 \rho \left(\frac{4}{3}\pi r_2^3\right) c \Delta T \\$Now, find the ratio $\frac{Q_1}{Q_2}: \\ \frac{Q_1}{Q_2} = \frac{3.375 \rho \left(\frac{4}{3}\pi r_2^3\right) c \Delta T}{\rho \left(\frac{4}{3}\pi r_2^3\right) c \Delta T} \\$Cancel out the common terms:$\\ \frac{Q_1}{Q_2} = 3.375 \\$Convert $3.375$ to a fraction:$\\ 3.375 = \frac{27}{8} \\$Therefore, the ratio of the heat required is $\frac{27}{8}. \\$This corresponds to Option 4.