Final answer:
To minimize the cost of constructing the energy drink container, we need to find the dimensions that will minimize the surface area. By using the volume formula and expressing the height in terms of the radius, we can derive the expression for the total cost in terms of the radius. Taking the derivative of the total cost with respect to the radius and setting it equal to zero will allow us to find the critical points and determine the dimensions that minimize the cost.
Step-by-step explanation:
To minimize the cost of constructing the energy drink container, we need to find the dimensions that will minimize the surface area. Let's denote the radius of the top and bottom of the cylinder as r and the height of the cylinder as h.
The volume of the cylinder is given by the formula V = πr²h. In this case, we have V = 18 fluid ounces = 18 * 1.80469 cubic inches = 32.48322 cubic inches.
The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side. Let's denote the cost per square inch of the lateral side as C.
The surface area of the top and bottom is given by the formula A_top_bottom = 2πr², and the surface area of the lateral side is given by the formula A_lateral = 2πrh.
Using the volume formula, we can express the height in terms of the radius: h = V / (πr²).
Substituting the expression for height in the formulas for the surface areas, we get:
A_top_bottom = 2πr²
A_lateral = 2πr * (V / (πr²)) = 2V / r
The total cost, C_total, is given by the formula C_total = (2C * A_top_bottom) + (C * A_lateral) = 2C * 2πr² + C * (2V / r) = 4Cπr² + 2CV / r.
To find the dimensions that minimize the cost, we need to find the critical points of C_total. Taking the derivative of C_total with respect to r and setting it equal to zero, we get:
8Cπr - 2CV / r² = 0
Simplifying the equation, we have:
8Cπr = 2CV / r²
4πr³ = V / C
Solving for r, we get:
r = (V / (4πC))^(1/3)
Substituting the values for V (32.48322 cubic inches) and C, we can calculate the radius, r = (32.48322 / (4 * 3.14159 * C))^(1/3).
Finally, substituting the calculated value of r back into the expression for height, we can find the height of the cylinder.