Question

a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will make 1938 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. 0.2 m 0.3 m 0.45 m

Explanation:

Step1: Calculate area of rectangular sides

There are 3 rectangular sides. Two with dimensions $0.3\times0.45$ and one with dimensions $0.2\times0.45$.
$A_{rect1}=2\times(0.3\times0.45) = 0.27$ $m^{2}$, $A_{rect2}=0.2\times0.45=0.09$ $m^{2}$, total rectangular area $A_{rect}=0.27 + 0.09=0.36$ $m^{2}$

Step2: Calculate area of half - cylinder top

The radius of the half - cylinder $r = 0.1$ m (since diameter is $0.2$ m), length of the cylinder is $l=0.45$ m. The area of the half - circular top $A_{semicircle}=\frac{1}{2}\times2\times\pi\times r\times l=\pi\times0.1\times0.45 = 0.1413$ $m^{2}$, and the area of the flat part of the half - cylinder (rectangular part) is $0.2\times0.45 = 0.09$ $m^{2}$. Total half - cylinder area $A_{cylinder}=0.1413+0.09 = 0.2313$ $m^{2}$

Step3: Calculate total surface area of one mailbox

$A = A_{rect}+A_{cylinder}=0.36 + 0.2313=0.5913$ $m^{2}$

Step4: Calculate total area for 1938 mailboxes

$A_{total}=1938\times0.5913=1938\times(0.5913)\approx1146$ $m^{2}$

Answer:

1146