(b) The surface area of a cuboid is \( 0,0292 \mathrm{~m}^{2} \) and its volume is \( 24 \mathrm{~cm}^{3} \). (1) Determine the surface area (in \( \mathrm{cm}^{2} \) ) and volume \( \left(\right. \) in \( \left.\mathrm{cm}^{3}\right) \) of the cuboid formed if the dimensions of the original cuboid are unultiplied by 5 . (2) Suppose that the volume of the original cuboid is increased by 8 times its value. What is the surface arei of the enlarged cuboid?

To solve the problem, we need to first convert the surface area of the cuboid from square meters to square centimeters, and then we can address the two parts of the question. ### Given: - Surface area of the cuboid: \( 0.0292 \, \text{m}^2 \) - Volume of the cuboid: \( 24 \, \text{cm}^3 \) ### Conversion of Surface Area: 1. Convert the surface area from square meters to square centimeters: \[ 0.0292 \, \text{m}^2 = 0.0292 \times 10,000 \, \text{cm}^2 = 292 \, \text{cm}^2 \] ### (1) Surface Area and Volume of the Enlarged Cuboid: If the dimensions of the original cuboid are multiplied by 5, the new surface area and volume can be calculated as follows: - **Surface Area**: The surface area of a cuboid scales with the square of the linear dimensions. Therefore, if the dimensions are multiplied by 5, the new surface area \( S' \) is: \[ S' = 5^2 \times S = 25 \times 292 \, \text{cm}^2 = 7300 \, \text{cm}^2 \] - **Volume**: The volume of a cuboid scales with the cube of the linear dimensions. Therefore, if the dimensions are multiplied by 5, the new volume \( V' \) is: \[ V' = 5^3 \times V = 125 \times 24 \, \text{cm}^3 = 3000 \, \text{cm}^3 \] ### (2) Surface Area of the Enlarged Cuboid with Increased Volume: If the volume of the original cuboid is increased by 8 times its value, the new volume \( V'' \) is: \[ V'' = 8 \times 24 \, \text{cm}^3 = 192 \, \text{cm}^3 \] To find the new surface area corresponding to this new volume, we need to determine the scale factor for the dimensions. The volume scales with the cube of the linear dimensions, so if the volume is increased by a factor of 8, the linear dimensions are multiplied by: \[ \text{Scale factor} = \sqrt[3]{8} = 2 \] Thus, the new surface area \( S'' \) can be calculated as: \[ S'' = 2^2 \times S = 4 \times 292 \, \text{cm}^2 = 1168 \, \text{cm}^2 \] ### Summary of Results: 1. The surface area of the cuboid formed by multiplying the dimensions by 5 is \( 7300 \, \text{cm}^2 \) and the volume is \( 3000 \, \text{cm}^3 \). 2. The surface area of the enlarged cuboid (with volume increased by 8 times) is \( 1168 \, \text{cm}^2 \).