Question

a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will make 1500 mailboxes this week. if each mailbox has dimensions as shown in the figure below, how many square meters of aluminum will be needed to make these mailboxes? in your calculations, use the value 3.14 for \\( \pi \\), and round up your answer to the next square meter.

Explanation:

Step1: Calculate the surface area of the rectangular part

The surface area of the rectangular part \(S_{1}\) has two faces of \(0.6\times0.7\), two faces of \(0.75\times0.7\), and one face of \(0.6\times0.75\).

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Step2: Calculate the surface area of the half - cylinder part

The diameter of the half - cylinder \(d = 0.6m\), so the radius \(r=\frac{d}{2}=0.3m\). The length of the half - cylinder (same as the length of the box) \(l = 0.75m\).
The lateral surface area of a full - cylinder is \(A = 2\pi rl\), for a half - cylinder, the lateral surface area \(S_{21}=\pi rl\), and the area of the circular face (since it's a half - cylinder, we have one circular face) \(S_{22}=\frac{1}{2}\pi r^{2}\) (but we don't need to consider the bottom circular face as it's attached to the box).

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Step3: Calculate the surface area of one mailbox

The surface area of one mailbox \(S=S_{1}+S_{2}\)

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Step4: Calculate the total surface area for 1500 mailboxes

The total surface area \(A_{total}=1500\times S\)

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Rounding up to the next square meter, \(A_{total}\approx4570\)

Answer:

\(4570\)