Question

a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will make 1298 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 π, and round up your answer to the next square meter.

Explanation:

Step1: Calculate the surface area of the mailbox

The mailbox consists of a rectangular part and a half - cylinder part.

• Rectangular part:

The dimensions of the rectangular part: length \(l = 2m\), width \(w=0.4m\).
The area of the two rectangular sides: \(2\times(2\times0.4)=1.6m^{2}\)

• Half - cylinder part:

The diameter of the half - cylinder \(d = 0.4m\), so the radius \(r=\frac{d}{2}=0.2m\)
The lateral surface area of a full - cylinder is \(A_{lateral}=2\pi r h\), for a half - cylinder, \(A_{half - lateral}=\pi r h\), where \(h = 2m\)
\(A_{half - lateral}=3.14\times0.2\times2 = 1.256m^{2}\)
The area of the circular end of the half - cylinder (since there is one circular end in the mailbox structure): \(A_{circle}=\pi r^{2}=3.14\times(0.2)^{2}=0.1256m^{2}\)

The total surface area of one mailbox \(A=1.6 + 1.256+0.1256=2.9816m^{2}\)

Step2: Calculate the total surface area for 1298 mailboxes

Multiply the surface area of one mailbox by the number of mailboxes.
\(Total\space A=1298\times2.9816\)
\(Total\space A = 1298\times(3 - 0.0184)=1298\times3-1298\times0.0184\)
\(1298\times3 = 3894\), \(1298\times0.0184=(1300 - 2)\times0.0184=1300\times0.0184-2\times0.0184=23.92 - 0.0368 = 23.8832\)
\(Total\space A=3894-23.8832 = 3870.1168\approx3870m^{2}\)

Answer:

\(3870\)