To calculate the total surface area of the aluminum required to manufacture the mailboxes, we need to consider the dimensions of a mailbox, which consists of a rectangular base and a half-cylinder top.
Assume the following dimensions for the mailbox:
- Height of the rectangular part = \( h \)
- Width of the rectangular base = \( w \)
- Depth of the rectangular base = \( d \)
- Radius of the half-cylinder top = \( r \)
The calculations involve two parts: the rectangular part and the half-cylinder part.
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Surface Area of the Rectangular Part: The surface area \( A_r \) of the rectangular part consists of:
- The front and back faces: \( 2 \times (h \times d) \)
- The two side faces: \( 2 \times (h \times w) \)
- The bottom face: \( w \times d \)
So, the total surface area of the rectangular part is: \[ A_r = 2(h \times d) + 2(h \times w) + (w \times d) \]
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Surface Area of the Half-Cylinder: The surface area \( A_c \) of the half-cylinder top consists of:
- The curved surface area of the half-cylinder: \( \pi \times r \times h \)
- The flat face (semicircle): \( \frac{1}{2} \pi r^2 \)
The curved surface area contributes: \[ A_c = \pi \times r \times d \]
Since we only consider the curved surface area (the flat base of the cylinder aligns with the top of the rectangular part), we can ignore the base.
Summing these areas gives us the total surface area \( A \): \[ A = A_r + A_c = (2(h \times d) + 2(h \times w) + (w \times d)) + (\pi r d) \]
To find out how many square meters of aluminum are needed for the production of \( n \) mailboxes: \[ \text{Total Surface Area for } n \text{ mailboxes} = n \cdot A \]
Example Calculation:
For example, if we have:
- \( n = 100 \) (mailboxes)
- \( h = 1 \) meter
- \( w = 0.5 \) meters
- \( d = 0.5 \) meters
- \( r = 0.25 \) meters
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Calculate Surface Area of One Mailbox: \[ A_r = 2(1 \times 0.5) + 2(1 \times 0.5) + (0.5 \times 0.5) = 1 + 1 + 0.25 = 2.25 , m^2 \] \[ A_c = \pi (0.25)(0.5) \approx 0.3927 , m^2 \] \[ A = 2.25 + 0.3927 \approx 2.6427 , m^2 \]
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Total Surface Area for 100 Mailboxes: \[ \text{Total Area} = 100 \times 2.6427 = 264.27 , m^2 \]
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Rounding Up: The final answer, rounded up, would be: \[ \lceil 264.27 \rceil = 265 , m^2 \]
Therefore, the company would need 265 square meters of aluminum to make 100 mailboxes. Adjust the calculation according to your specific dimensions as needed.