Two pyramids, \( A \) and \( B \), are mathematically similar. Pyramid \( A \) has surface area \( 12 \mathrm{~cm}^{2} \) and volume \( 8 \mathrm{~cm}^{3} \). Pyramid \( B \) has surface area \( 75 \mathrm{~cm}^{2} \) . Work out the volume of pyramid \( B \).

To find the volume of pyramid \( B \), we can use the properties of similar figures. Since the pyramids are similar, the ratios of their corresponding surface areas and volumes will be equal to the square and cube of the ratio of their corresponding linear dimensions, respectively. Let the scale factor (the ratio of corresponding lengths) be \( k \). The ratio of the surface areas is given by: \[ \frac{\text{Surface Area of } B}{\text{Surface Area of } A} = \frac{75 \, \text{cm}^2}{12 \, \text{cm}^2} = \frac{75}{12} = \frac{25}{4} \] This means: \[ k^2 = \frac{25}{4} \] To find \( k \), we take the square root: \[ k = \sqrt{\frac{25}{4}} = \frac{5}{2} \] Now, to find the volume of pyramid \( B \), we use the relationship for volumes: \[ \frac{\text{Volume of } B}{\text{Volume of } A} = k^3 \] First, let's calculate \( k^3 \): \[ k^3 = \left(\frac{5}{2}\right)^3 = \frac{125}{8} \] Now we can use the volume of pyramid \( A \) to find the volume of pyramid \( B \): \[ \frac{\text{Volume of } B}{8 \, \text{cm}^3} = \frac{125}{8} \] Let \( V_B \) be the volume of pyramid \( B \): \[ V_B = 8 \, \text{cm}^3 \times \frac{125}{8} = 125 \, \text{cm}^3 \] Thus, the volume of pyramid \( B \) is: \[ \boxed{125 \, \text{cm}^3} \]