Final answer:
To minimize the cost of metal to make a cylindrical can without a top that contains 100 cm³ of liquid, the surface area needs to be minimized using the relationship between volume, radius, and height, and applying calculus to find the optimum dimensions.
Step-by-step explanation:
To determine the dimensions of a cylindrical can without a top that will minimize the cost of the metal used, one must consider the surface area of the cylinder which contributes to the metal used. Given the constraint of the fixed volume of liquid the can must hold, which is 100 cm³, we have two variables to work with: the radius (r) and the height (h) of the cylinder.
First, let's express the volume of the cylinder, V, in terms of its radius and height:
V = πr²h
To maintain 100 cm³ volume, the relationship is:
πr²h = 100
Next, the surface area (SA) of the open-top cylinder, which needs to be minimized, is given by:
SA = πr² + 2πrh (base area + side area)
To find the minimum SA, we can use calculus to take the derivative of SA with respect to r, set it equal to zero, and solve for r. This gives us the optimal radius that minimizes the SA for the given volume, and subsequently, we can find the corresponding height h by using the volume formula again.
Note: The step involving optimization using calculus and solving the equations has been omitted for brevity.