Armin is designing a bottle for a new
shampoo. The bottle should have a volume
of 450 cm3. Armin wants to minimize the
amount of plastic used in the bottle.
Determine the dimensions of the bottle
with the minimum surface area in each
case. Explain your method.
a) The bottle is a cylinder.
b) The bottle is a triangular prism with a
base that is an equilateral triangle.
To determine the dimensions of the bottle with the minimum surface area for each case, we will use the concept of optimization. The goal is to minimize the surface area while maintaining a volume of 450 cm³.
(a) The bottle is a cylinder:
Let's use the formula for the surface area of a cylinder to express it in terms of its dimensions.
The surface area (A) of a cylinder is given by:
A = 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder.
However, we are given that the volume (V) of the cylinder is 450 cm³, which can be expressed as:
V = πr²h
We need to find the dimensions (r and h) that minimize the surface area (A) while simultaneously satisfying the volume constraint (V = 450 cm³).
To minimize A, we can use the volume equation to express h in terms of r:
h = V / (πr²)
Now, substitute this value of h in the surface area formula:
A = 2πr² + 2πr(V / (πr²))
A = 2πr² + 2V/r
Now, we can differentiate A with respect to r and set it equal to zero to find the minimum surface area. Let's do that:
dA/dr = 4πr - 2V/r²
Set dA/dr = 0 and solve for r:
4πr - 2V/r² = 0
4πr = 2V/r²
2r³ = V/π
r³ = V / (2π)
r = ³√(V / (2π))
Now, substitute this value of r back into the volume equation to find h:
h = V / (πr²)
h = V / (π(³√(V / (2π)))²)
So, the dimensions of the cylinder with the minimum surface area are:
Radius (r) = ³√(V / (2π))
Height (h) = V / (π(³√(V / (2π)))²)
Substituting V = 450 cm³, we can calculate the actual values of r and h.
(b) The bottle is a triangular prism with a base that is an equilateral triangle:
For this case, we need to minimize the surface area of the triangular prism while maintaining the volume of 450 cm³.
The surface area (A) of a triangular prism is given by:
A = 2B + Ph, where B is the area of the base, P is the perimeter of the base, and h is the height of the prism.
In an equilateral triangle, the base, B, is given by:
B = (√3 / 4) * s², where s is the length of the side of the equilateral triangle.
We are also given that the volume (V) of the prism is 450 cm³, and for an equilateral triangle, the height (h) is related to the side length (s) by:
h = (2V) / (s * √3)
Substituting these values into the surface area formula, we have:
A = 2((√3 / 4) * s²) + Ps = (√3 / 2) * s² + Ps
Now, differentiate A with respect to s and set it equal to zero to find the minimum surface area:
dA/ds = (√3 / 2) * 2s + P = √3s + P
Set dA/ds = 0 and solve for s:
√3s + P = 0
√3s = -P
s = -P / √3
Since the side length of the triangle cannot be negative, we can disregard this result. Instead, we can use the fact that the perimeter of an equilateral triangle is three times the length of one side (P = 3s).
Substituting P = 3s into the equation:
√3s + P = √3s + 3s = 4s
Set this equal to zero and solve for s:
4s = 0
s = 0
Although this solution gives us zero as the side length, it is not physically meaningful. We can conclude that in this case, it is not possible to minimize the surface area while maintaining a volume of 450 cm³ for a triangular prism with an equilateral base.
In summary:
(a) The dimensions of the cylinder with the minimum surface area can be found using:
Radius (r) = ³√(V / (2π))
Height (h) = V / (π(³√(V / (2π)))²)
(b) It is not possible to minimize the surface area while maintaining a volume of 450 cm³ for a triangular prism with an equilateral base.