To find the total surface area of the aluminum mailboxes, we first need to calculate the surface area of one mailbox which consists of a rectangular box and a half-cylinder on top.
Step 1: Calculate the Surface Area of One Mailbox
Dimensions of the Mailbox:
- Height of the rectangular part (box): \( h = 0.45 , \text{m} \)
- Width of the box (this will also be the diameter of the half-cylinder): \( d = 0.2 , \text{m} \) → The radius \( r = \frac{d}{2} = 0.1 , \text{m} \)
- Length of the box: \( L = 0.5 , \text{m} \)
Area of the Rectangular Box:
The rectangular box has:
- Two sides of dimensions \( L \times h \)
- Two ends of dimensions \( d \times h \) (where \( d \) is the width)
- The bottom is not included as it is open
Calculating the areas:
- Area of the two long sides:
\[ 2 \times (L \times h) = 2 \times (0.5 , \text{m} \times 0.45 , \text{m}) = 0.45 , \text{m}^2 \] - Area of the two ends:
\[ 2 \times (d \times h) = 2 \times (0.2 , \text{m} \times 0.45 , \text{m}) = 0.18 , \text{m}^2 \]
Total area of the rectangular part (excluding the bottom): \[ \text{Area of box} = 0.45 + 0.18 = 0.63 , \text{m}^2 \]
Area of the Half-Cylinder:
The curved surface area of a cylinder is given by the formula \( 2 \pi r h \). Since we only have a half-cylinder, it will be: \[ \text{Half-cylinder area} = \text{Curved surface area of cylinder} = \pi r L \]
Substituting the values: \[ \text{Half-cylinder area} = 3.14 \times 0.1 , \text{m} \times 0.5 , \text{m} = 0.157 , \text{m}^2 \]
Total Surface Area of One Mailbox:
The total surface area of one mailbox: \[ \text{Total Area} = \text{Area of box} + \text{Half-cylinder area} \] \[ \text{Total Area} = 0.63 , \text{m}^2 + 0.157 , \text{m}^2 = 0.787 , \text{m}^2 \]
Step 2: Calculate Total Surface Area for 1646 Mailboxes
Now calculate the total surface area for 1646 mailboxes: \[ \text{Total Area for 1646 mailboxes} = 0.787 , \text{m}^2 \times 1646 = 1290.282 , \text{m}^2 \]
Step 3: Round Up to the Next Square Meter
Rounding up \( 1290.282 , \text{m}^2 \) gives: \[ 1291 , \text{m}^2 \]
Conclusion
The total area of aluminum needed to make 1646 mailboxes is 1291 square meters.