To calculate the total surface area of a tower consisting of a square pyramid on top of a rectangular prism, we need to find the surface areas of both shapes and then combine them, being careful not to double count the area where the pyramid sits on top of the prism.
Step 1: Understanding the dimensions
Rectangular Prism:
- Let's denote:
- Length \( l \)
- Width \( w \)
- Height \( h \)
Square Pyramid:
- Base side length \( b \)
- Height \( H \)
Step 2: Surface Area Calculations
-
Surface Area of the Rectangular Prism: The surface area of a rectangular prism is calculated using the formula: \[ \text{Surface Area}_{prism} = 2(lw + lh + wh) \]
-
Surface Area of the Square Pyramid: The surface area of a square pyramid (including the base) is: \[ \text{Surface Area}_{pyramid} = b^2 + 2 \cdot b \cdot s \] where \( s \) is the slant height of the pyramid. The slant height can be calculated using the Pythagorean theorem if the perpendicular height \( H \) is known: \[ s = \sqrt{(\frac{b}{2})^2 + H^2} \]
Step 3: Combine Surface Areas
- When placing the pyramid on top of the prism, the area of the base of the pyramid will not be counted in the total surface area since that face is not exposed.
- Thus, the total surface area of the tower becomes: \[ \text{Total Surface Area} = \text{Surface Area}{prism} + \text{Surface Area}{pyramid} - b^2 \]
Example Calculation
Let's consider an example with arbitrary values:
- Assume the prism has dimensions: \( l = 10 \) cm, \( w = 5 \) cm, \( h = 15 \) cm.
- Assume the pyramid has a base side length \( b = 5 \) cm and a height \( H = 10 \) cm.
Surface Area of the Rectangular Prism
\[ \text{Surface Area}_{prism} = 2(10 \cdot 5 + 10 \cdot 15 + 5 \cdot 15) = 2(50 + 150 + 75) = 2(275) = 550 \text{ square cm} \]
Surface Area of the Square Pyramid
Calculate the slant height \( s \): \[ s = \sqrt{\left(\frac{5}{2}\right)^2 + 10^2} = \sqrt{6.25 + 100} = \sqrt{106.25} \approx 10.3 \text{ cm} \] Then the surface area of the pyramid: \[ \text{Surface Area}_{pyramid} = 5^2 + 2 \cdot 5 \cdot 10.3 = 25 + 103 = 128 \text{ square cm} \]
Total Surface Area
Combine the areas: \[ \text{Total Surface Area} = 550 + 128 - 25 = 653 \text{ square cm} \]
Conclusion
You would follow a similar calculation depending on the specific dimensions of the tower. However, as the options given are numerical answers, we need specific dimensions to find the correct total surface area. Therefore, if you have specific dimensions for the pyramid and prism, please provide them for an accurate calculation. Without the dimensions provided in your question, I can't determine the exact total surface area or match it with one of the options given.