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 rectangular part

The rectangular part has two faces with dimensions \(0.3\times0.65\), two faces with dimensions \(0.4\times0.65\), and one face with dimensions \(0.3\times0.4\).
The area of the two \(0.3\times0.65\) faces is \(2\times(0.3\times0.65)= 0.39\)
The area of the two \(0.4\times0.65\) faces is \(2\times(0.4\times0.65)=0.52\)
The area of the \(0.3\times0.4\) face is \(0.3\times0.4 = 0.12\)
The total area of the rectangular part is \(0.39+0.52 + 0.12=1.03\)

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

The formula for the lateral surface area of a full - cylinder is \(A = 2\pi r h\), for a half - cylinder, it is \(A=\pi r h\). The diameter of the half - cylinder is \(0.4m\), so the radius \(r=\frac{0.4}{2}=0.2m\), and the height \(h = 0.65m\). The lateral surface area of the half - cylinder is \(3.14\times0.2\times0.65=0.4082\)
The area of the circular end (since it is a half - cylinder, we consider one - half of the area of a full - circle). The area of a full - circle is \(\pi r^{2}\), for a half - circle, it is \(\frac{1}{2}\pi r^{2}\). So \(\frac{1}{2}\times3.14\times(0.2)^{2}=0.0628\)

Step3: Calculate the total surface area of one mailbox

The total surface area of one mailbox is \(1.03+0.4082+0.0628 = 1.501\)

Step4: Calculate the total surface area for 1298 mailboxes

Multiply the surface area of one mailbox by the number of mailboxes: \(1.501\times1298\)
\(1.501\times1298=(1.5 + 0.001)\times1298=1.5\times1298+0.001\times1298\)
\(1.5\times1298 = 1947\), \(0.001\times1298=1.298\)
\(1947+1.298 = 1948.298\approx1949\)

Answer:

\(1949\)