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

Explanation:

Step1: Calculate surface - area of the box part

The box has 5 faces (the bottom, two sides, and two ends). The area of the bottom is $A_{bottom}=0.4\times0.6 = 0.24$ m². The area of the two side - faces is $A_{sides}=2\times(0.55\times0.6)=0.66$ m². The area of the two end - faces is $A_{ends}=2\times(0.55\times0.4)=0.44$ m². The total surface area of the box part is $A_{box}=0.24 + 0.66+0.44=1.34$ m².

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

The length of the half - cylinder is $l = 0.6$ m, and the radius $r=\frac{0.4}{2}=0.2$ m. The curved surface area of the half - cylinder is $A_{curved}=\frac{1}{2}\times2\pi rl=\pi rl$. Substituting $r = 0.2$ m and $l = 0.6$ m, we get $A_{curved}=3.14\times0.2\times0.6 = 0.3768$ m². The area of the two semi - circular ends (which together form a full circle) is $A_{circle}=\pi r^{2}=3.14\times0.2^{2}=0.1256$ m². The total surface area of the half - cylinder part is $A_{cylinder}=0.3768 + 0.1256=0.5024$ m².

Step3: Calculate surface - area of one mailbox

The surface area of one mailbox is $A = A_{box}+A_{cylinder}=1.34 + 0.5024=1.8424$ m².

Step4: Calculate surface - area of 1863 mailboxes

The total surface area of 1863 mailboxes is $A_{total}=1863\times1.8424$. $A_{total}=1863\times1.8424 = 3432.4912$ m². Rounding up to the next square meter, $A_{total}=3433$ m².

Answer:

3433