a. In order to determine how much soap material is used for each bar of soap, consider that the shape of the bar is a cylinder, then, use the folowing formula for the volume of a cylinder:
V = π r² h
where r is the radius of the base and h is the height.
The diameter is 5 cm, then, the radius is r = d/2 = 5 cm/2 = 2.5 cm
The height is h = 2 cm
replace the previous values into the formula for V:
V = (3.14)(2.5 cm)²(2 cm)
V = 39.25 cm³
Hence, 39.25 cm³ of soap material is used to make each bar.
b. In this case it is necessary to calculate the suface area of the plastic used. As before, take into account that the bar has a cylindrical shape. Then, use the following formula for the surface area of a cylinder:
S = 2πr² + 2πrh
replace the values of the parameters to find S:
S = 2π(2.5 cm)² + 2π(2.5cm)(2cm)
S = 70.68 cm²
Hence, 70.68 cm² of plastic is required to wrap each bar
c. Take into account that the soap must be stacked no more than 5 bars high, and that the company wants to ship a minimum of 60 bars.
If there are 5 bars high x 3 bars length, in one face of the box you have 15 bars. Moreover, if there are 4 bars width, then, you have a total of 15x4 = 60 bars in one box, which is what the company wants.
The height of the box is given by 5 times the height of one bar (because there are 5 bars high). The length is given by 3 times the diameter of one bar and the width of the box is given by 4 times the diameter of one bar:
height of the box = 5(2 cm) = 10 cm
length of the box = 3(5 cm) = 15 cm
width of the box = 4(5 cm) = 20 cm
d. First, calculate the volume of the box, as follow:
V = h x l x w = (10 cm)(15 cm)(20 cm) = 3,000 cm³
take into account the conversion 1,000 cm³ = 1 L, then, the capacity of the box is:
V = 3,000 cm³ = 3L