Question

A company manufactures aluminum mailboxes in the shape of a box with a half-cylinder top. The company will make 1728 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 x, and round up your answer to the next square meter.
?

Answer 1

We can start by calculating the total surface area of each mailbox, which includes the surface area of the box and the surface area of the half-cylinder top.

The dimensions of the mailbox are given in the figure, with height h = 40 cm, length l = 60 cm, and width w = 30 cm. The radius of the half-cylinder top is also given as r = 15 cm.

The surface area of the box can be calculated as follows:

Front and back faces: 2lw = 2(60 cm)(40 cm) = 4800 cm^2

Top and bottom faces: 2wh = 2(30 cm)(40 cm) = 2400 cm^2

Side faces: 2lh = 2(60 cm)(40 cm) = 4800 cm^2

Total surface area of the box = 4800 cm^2 + 2400 cm^2 + 4800 cm^2 = 12000 cm^2

The surface area of the half-cylinder top can be calculated as:

Curved surface area: πrh = 3.14(15 cm)(40 cm) = 1884 cm^2

Circular top and bottom faces: 2πr^2 = 2(3.14)(15 cm)^2 = 1413 cm^2

Total surface area of the half-cylinder top = 1884 cm^2 + 1413 cm^2 = 3297 cm^2

Therefore, the total surface area of each mailbox is:

Total surface area = surface area of box + surface area of half-cylinder top

= 12000 cm^2 + 3297 cm^2

= 15297 cm^2

To find the total surface area of 1728 mailboxes, we can multiply the surface area of one mailbox by the number of mailboxes:

Total surface area of 1728 mailboxes = 1728 mailboxes x 15297 cm^2 per mailbox

= 26,431,616 cm^2

Converting this to square meters, we get:

Total surface area of 1728 mailboxes = 264,316.16 cm^2 = 26.43 m^2 (rounded up)

Therefore, approximately 27 square meters of aluminum will be needed to make these mailboxes.

I Hope This Helps!

Answer 2

2145 m^2

Explanation:

The area of one mailbox = the area of a rectangular box - the top of the box plus half the area of a cylinder.

SA = SA(box) - SA(top) + ½SA(cylinder)

1. Surface area of box

The formula for the surface area of a rectangular box is

SA = 2(lw + lh + wh)

Data:

l = 0.55 m

w = 0.3 m

h = 0.4 m

Calculations:

2(Top + Bottom = 2lw = 2 × 0.55 × 0.3 = 0.33 m²

2(Left + Right) = 2wh = 2 × 0.55 × 0.4 = 0.44 m²

2(Front + Back) = 2lh = 2 × 0.3 × 0.4 = 0.24 m²

Total area = 1.01 m²

2. Surface area of cylinder

The formula for the surface area of a cylinder is

SA = A(top) + A (base) + A(side) = 2A(base) + A(side)

Data:

d = 0.3 m

h = 0.55 m

Calculations:

r = ½d = ½ × 0.3 = 0.15 m

3. Excluded area

1 top = ½ × 0.33 m² = 0.165 m²

½ cylinder = ½ × 0.6594 m² = 0.3297 m²

Total excluded = 0.4947 m²

4. Surface area of 1 mailbox

SA = (1.01 + 0.6594 - 0.4927) m² = 1.1767 m²

5. Total area of 1823 mailboxes