Question

a company manufactures aluminum mailboxes in the shape of a box with a half - cylinder top. the company will make 1863 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. 0.55 m 0.4 m 0.6 m

Explanation:

Step1: Calculate area of half - cylinder top

The length of the half - cylinder is the length of the box $l = 0.6$m, and the diameter of the half - cylinder $d=0.4$m, so the radius $r = 0.2$m. The lateral surface area of a half - cylinder is $A_{1}=\frac{1}{2}\times2\pi rl=\pi rl$. Substituting $r = 0.2$m and $l = 0.6$m and $\pi=3.14$, we get $A_{1}=3.14\times0.2\times0.6 = 0.3768$m².

Step2: Calculate area of the two ends of the half - cylinder

The two ends of the half - cylinder together form a full - circle. The area of a circle is $A_{2}=\pi r^{2}$. Substituting $r = 0.2$m and $\pi = 3.14$, we get $A_{2}=3.14\times0.2^{2}=0.1256$m².

Step3: Calculate area of the rectangular sides of the box

There are two rectangular sides with dimensions $0.6$m×$0.55$m and two with dimensions $0.4$m×$0.55$m. The area of the sides is $A_{3}=2\times(0.6\times0.55 + 0.4\times0.55)=2\times0.55\times(0.6 + 0.4)=1.1$m².

Step4: Calculate area of the bottom of the box

The area of the bottom of the box is $A_{4}=0.6\times0.4 = 0.24$m².

Step5: Calculate total surface area of one mailbox

The total surface area of one mailbox $A=A_{1}+A_{2}+A_{3}+A_{4}=0.3768+0.1256 + 1.1+0.24=1.8424$m².

Step6: Calculate total surface area of 1863 mailboxes

The total surface area of 1863 mailboxes is $1863\times1.8424\approx3432.49$m². Rounding up to the next square meter, we get 3433 m².

Answer:

3433