To calculate the surface area of the composite solid formed by the two rectangular prisms, we first need to find the surface areas of both prisms separately and then account for the area that is not exposed where they are joined.
Step 1: Calculate the Surface Area of Each Prism
1. Smaller Rectangular Prism:
- Dimensions:
- Length \( L = 10 \) m
- Width \( W = 2 \) m
- Height \( H = 9 \) m
The surface area \( A \) of a rectangular prism is given by the formula: \[ A = 2(LW + LH + WH) \]
Thus, for the smaller prism: \[ A_s = 2(10 \times 2 + 10 \times 9 + 2 \times 9) \] Calculating each term:
- \( 10 \times 2 = 20 \)
- \( 10 \times 9 = 90 \)
- \( 2 \times 9 = 18 \)
Adding these: \[ A_s = 2(20 + 90 + 18) = 2(128) = 256 \text{ m}^2 \]
2. Bigger Rectangular Prism:
- Dimensions:
- Length \( L = 32 \) m
- Width \( W = 17 \) m
- Height \( H = 25 \) m
For the bigger prism: \[ A_b = 2(32 \times 17 + 32 \times 25 + 17 \times 25) \] Calculating each term:
- \( 32 \times 17 = 544 \)
- \( 32 \times 25 = 800 \)
- \( 17 \times 25 = 425 \)
Adding these: \[ A_b = 2(544 + 800 + 425) = 2(1769) = 3538 \text{ m}^2 \]
Step 2: Find the Area of Overlap
The overlapping area needs to be subtracted. This area depends on the size of the base of the smaller prism that is in contact with the larger prism. We assume the smaller prism sits completely within the larger one.
Assuming the smaller prism is sitting on a face of the larger prism, the base of the smaller prism (10 m x 2 m) will be in contact. Therefore: \[ \text{Area of overlap} = 10 \times 2 = 20 \text{ m}^2 \]
Step 3: Calculate the Total Exposed Surface Area
Now we can compute the total surface area of the composite solid: \[ \text{Total Surface Area} = A_s + A_b - \text{Area of overlap} \] \[ = 256 + 3538 - 20 = 3774 \text{ m}^2 \]
Step 4: Convert Square Meters to Square Millimeters
1 square meter (m²) = 1,000,000 square millimeters (mm²), so: \[ 3774 \text{ m}^2 = 3774 \times 1,000,000 = 3,774,000,000 \text{ mm}^2 \]
Conclusion
With the conversion, the final answer is: \[ \text{Total Surface Area} = 3,774,000,000 \text{ mm}^2 \] Since the options provided do not match the calculated surface area or seem to lack additional context about where measurements might have been misinterpreted, neither option, \(3614 \text{ mm}^2\) or \(3704 \text{ mm}^2\), is valid based on standard calculations. Thus, based on calculations, the correct answer is not listed in the multiple-choice answers.
For an accurate context of question and numeric form, please ensure accurate dimensions and overlaps shall clarify further.