Make the conversion from fluid ounces to cubic centimeters. Pick a shape and look for the formula to find its volume. Fix one dimension and find the other(s) so that the container holds 8 fluid ounces.
B
b
Look for the formulas to find the surface area of the shapes used in Part A. The lower the ratio, lesser the amount of material needed to make the container. Did you try a spherical container?
A
a
See solution.
B
b
See solution.
Practice makes perfect
a
We will begin by making the conversion from 8 fluid ounces to cubic centimeters.
8 fl oz * 29.573 53 cm^3/1 fl oz = 236.588 24 cm^3
This will make the construction of each container easier because we will define each dimension in centimeters. For our containers, we will consider a cylinder, a prism, and a sphere.
Volume Formulas
Cylinder
V_c = π r^2h
Prism
V_p = Bh
Sphere
V_s = 4/3Ï€ r^3
From these formulas, we will get the respective dimensions for each solid.
Cylindrical Container
Let's consider a cylindrical container such that the radius of the base is r=2 cm. Recall we want the volume for the container to be 236.588 24 cm^3.
Ï€ r^2h = 236.588 24 cm^3
Let's substitute r=2 into the equation above and solve it for h.
In consequence, a cylindrical container with a radius of 2 cm and a height of 18.827 094 cm holds a volume of 8 fluid ounces.
Square Prism Container
Let's consider a square of side length 5 cm as the base of the prism. With these dimensions, we have that the base area is 25 cm^2.
25* h = 236.588 24 cm^3
By solving the equation above for h, we will get h=9.463 529 cm. Below, we draw a container with these dimensions, which holds 8 fluid ounces.
Spherical Container
For the third container, we will consider a sphere. By equating the formula to find its volume and 236.588 24 cm^3, we will be able to find the radius of our container.
4/3Ï€ r^3 = 236.588 24 cm^3
⇒
r = 3.836 79 cm
Below, we draw our third container which is a sphere of radius r = 3.836 79 cm.
Note that the three containers mentioned above are just examples. There are other containers with different dimensions that also hold the same volume.
b
We need to calculate the surface area of each of the containers we sketched in Part A. So, let's begin by writing the formulas to find the surface area of each solid.
Surface Area Formulas
Cylinder
S_c = 2Ï€ rh + 2Ï€ r^2
Prism
S_p = Ph+2B
Sphere
S_s = 4Ï€ r^2
To find the surface area of our cylindrical container, we must substitute r=2 cm and h=18.827 094 cm into the first formula.
Consequently, the surface area of the cylindrical container is S_c = 261.720 98 cm^2. Doing a similar process we can find the surface area of the remaining two containers. We summarize the computations in the table below.
Container
Dimensions (cm)
Formula
Surface Area (cm^2)
Cylinder
l r=2 h=18.827 094
S_c = 2Ï€ rh + 2Ï€ r^2
S_c = 261.720 98
Prism
l w=5 l=5 h=9.463 529
S_p = Ph+2B
S_p = 239.270 58
Sphere
r=3.836 79
S_s = 4Ï€ r^2
S_s = 184.989
Next, let's write and simplify the ratio of the surface area to the volume of each solid. Remember that the volume of the three solids is V=8 fl oz.
Surface Area / Volume (cm^2 per fl oz)
S_c/V=261.720 98/8 = 32.715 12
S_p/V=239.270 58/8 = 29.908 82
S_s/V=184.989/8 = 23.123 62
As we can see, the third ratio is the minimum one. This tells us that, for an 8 fluid ounces container, the spherical container can be made for the lowest material cost. In fact, among all the solids we know so far, the sphere is the one that minimizes the ratio.
Least Amount of Material:
Spherical Container
If the cost of producing a container depends only on the amount of material, we can say that the spherical container is the cheapest to produce (here we are ignoring the machinery needed to make each type of container).
