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 in each case, we need to apply basic mathematical concepts and formulas.
a) For a cylindrical bottle:
Let the radius of the base of the cylinder be r and the height of the cylinder be h.
To find the minimum surface area, we need to consider two factors: the volume constraint and the objective to minimize the surface area.
The volume of a cylinder is given by the formula V = πr^2h. Here, we are given the volume as 450 cm^3. So we can write the equation as:
πr^2h = 450
To minimize the surface area, we need to find a relationship between the variables r and h to optimize it. The surface area of a cylinder is given by the formula SA = 2πr^2 + 2πrh.
To solve this problem, we use the method of Lagrange multipliers. The objective function for minimizing the surface area is:
F(r, h) = 2πr^2 + 2πrh
We introduce a constraint function g(r, h) = πr^2h - 450.
Now, we form the Lagrangian function:
L(r, h, λ) = F(r, h) - λg(r, h)
Taking the partial derivative of L with respect to r, h, and λ, and setting them equal to zero, we can solve for r, h, and λ.
The resulting values of r, h, and λ will give the dimensions of the bottle with the minimum surface area.
b) For a triangular prism bottle:
Let the side length of the equilateral triangle base be a, and the height of the triangular prism be h.
The volume of a triangular prism is given by the formula V = (1/4)*√3*a^2*h. Again, we are given the volume as 450 cm^3. So we can write the equation as:
(1/4)*√3*a^2*h = 450
Similar to the previous case, to minimize the surface area, we need to find a relationship between the variables a and h to optimize it. The surface area of a triangular prism is given by the formula SA = a^2 + 3ah.
We can apply a similar approach with Lagrange multipliers. The objective function for minimizing the surface area is:
F(a, h) = a^2 + 3ah
We introduce the constraint function g(a, h) = (1/4)*√3*a^2*h - 450.
Using the Lagrangian function L(a, h, λ), taking the partial derivative of L with respect to a, h, and λ, and setting them equal to zero, we can solve for a, h, and λ.
The resulting values of a, h, and λ will give the dimensions of the bottle with the minimum surface area.