a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will…

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.

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.

Answer

Explanation:

Step1: Calculate the surface area of the rectangular part

The rectangular part has two faces with dimensions (0.4\times0.65), two faces with dimensions (0.3\times0.65), and one face with dimensions (0.3\times0.4). [ \begin{align*} S_{1}&=2\times(0.4\times0.65)+2\times(0.3\times0.65)+0.3\times0.4\ &=2\times0.26 + 2\times0.195+0.12\ &=0.52+0.39 + 0.12\ &=1.03 \end{align*} ]

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

The diameter of the half - cylinder (d = 0.4m), so the radius (r=\frac{d}{2}=0.2m). The length of the half - cylinder (same as the height of the mailbox) (h = 0.65m). The lateral surface area of the half - cylinder (S_{lateral}=\pi rh), and the area of the circular end (S_{end}=\frac{1}{2}\pi r^{2}) [ \begin{align*} S_{2}&=\pi rh+\frac{1}{2}\pi r^{2}\ &=3.14\times0.2\times0.65+\frac{1}{2}\times3.14\times(0.2)^{2}\ &=3.14\times(0.13 + 0.02)\ &=3.14\times0.15\ &=0.471 \end{align*} ]

Step3: Calculate the total surface area of one mailbox

(S = S_{1}+S_{2}=1.03 + 0.471=1.501)

Step4: Calculate the total surface area for 1298 mailboxes

(S_{total}=1.501\times1298\approx1949)

Answer:

1949