The areas of two plates have a ratio of \( 9: 16 \). The smaller plate has an area of \( 36 \pi \) square inches. a. What is the ratio of the radius of the smaller plate to the radius of the larger plate? b. What is the radius of the larger plate?

To find the radius ratio, we know that the area \( A \) of a circle is given by \( A = \pi r^2 \). Given that the area ratio of the two plates is \( 9:16 \), we can express the areas as \( A_1 = 9k \) and \( A_2 = 16k \) for some constant \( k \). The smaller plate's area is \( 36\pi \), corresponding to \( A_1 = 36\pi \). Thus, we set up the equation: \[ 9k = 36\pi \implies k = 4\pi \] Now, for the larger plate's area: \[ A_2 = 16k = 16(4\pi) = 64\pi \] Next, we find the radii. Using the area formula, we find the radius of the smaller plate: \[ A_1 = \pi r_1^2 \implies 36\pi = \pi r_1^2 \implies r_1^2 = 36 \implies r_1 = 6 \] And for the larger plate: \[ A_2 = \pi r_2^2 \implies 64\pi = \pi r_2^2 \implies r_2^2 = 64 \implies r_2 = 8 \] Now we can determine the ratio of the radii: \[ \frac{r_1}{r_2} = \frac{6}{8} = \frac{3}{4} \] Thus, the ratio of the radius of the smaller plate to the radius of the larger plate is \( \frac{3}{4} \) (part a). For part (b), the radius of the larger plate has been calculated as \( 8 \) inches. In summary: a. The ratio of the radius of the smaller plate to the radius of the larger plate is \( \frac{3}{4} \). b. The radius of the larger plate is \( 8 \) inches.