Question

a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will make 1634 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 area of half - cylinder top

The radius of the half - cylinder $r=\frac{0.5}{2}=0.25$ m and length $l = 0.75$ m. The lateral surface area of a half - cylinder is $A_{1}=\frac{1}{2}\times2\pi rl=\pi rl$. Substituting values, we get $A_{1}=3.14\times0.25\times0.75 = 0.58875$ m². The area of the circular end of the half - cylinder is $A_{2}=\frac{1}{2}\pi r^{2}=\frac{1}{2}\times3.14\times(0.25)^{2}= 0.098125$ m².

Step2: Calculate area of box part

The box has two sides with dimensions $0.75\times0.7$ m, area of these two sides $A_{3}=2\times0.75\times0.7 = 1.05$ m². It has two sides with dimensions $0.5\times0.7$ m, area of these two sides $A_{4}=2\times0.5\times0.7 = 0.7$ m², and one bottom with dimensions $0.5\times0.75$ m, area $A_{5}=0.5\times0.75 = 0.375$ m².

Step3: Calculate surface area of one mailbox

The surface area of one mailbox $A = A_{1}+A_{2}+A_{3}+A_{4}+A_{5}=0.58875 + 0.098125+1.05+0.7+0.375=2.811875$ m².

Step4: Calculate surface area of 1634 mailboxes

The total surface area for 1634 mailboxes is $1634\times2.811875 = 4594.60375$ m². Rounding up, we get 4595 m².

Answer:

4595