(b) Marianne sells two sizes of photo. These photos are mathematically simlar rectangles. The smaller photo has length \( 15 \mathrm{~cm}^{2} \) and width 12 cm The larger photo has area \( 352.8 \mathrm{~cm}^{2} \) Calculate the length of the larger photo.

To find the length of the larger photo, we start by calculating the area of the smaller photo. The area is given by multiplying the length by the width: \[ \text{Area of smaller photo} = 15 \, \text{cm} \times 12 \, \text{cm} = 180 \, \text{cm}^2. \] Since the two photos are mathematically similar, the ratio of the areas will equal the square of the ratio of their corresponding lengths. Let the length of the larger photo be \( l \) cm and the width be \( w \) cm. The area of the larger photo is given as \( 352.8 \, \text{cm}^2 \). The ratio of the areas can be expressed as: \[ \frac{352.8}{180} = \left(\frac{l}{15}\right)^2. \] Calculating the ratio: \[ \frac{352.8}{180} = 1.96. \] Now we solve for the ratio of the lengths: \[ \left(\frac{l}{15}\right)^2 = 1.96 \implies \frac{l}{15} = \sqrt{1.96}. \] Taking the square root gives: \[ \frac{l}{15} = 1.4 \implies l = 15 \times 1.4 = 21 \, \text{cm}. \] Thus, the length of the larger photo is \( 21 \, \text{cm} \).