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Step-by-step explanation:
<u>Optimizing With Derivatives </u>
The procedure to optimize a function (find its maximum or minimum) consists in :
- Produce a function which depends on only one variable
- Compute the first derivative and set it equal to 0
- Find the values for the variable, called critical points
- Compute the second derivative
- Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum
We know a cylinder has a volume of 4 . The volume of a cylinder is given by
Equating it to 4
Let's solve for h
A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is
Replacing the formula of h
Simplifying
We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero
Rearranging
Solving for r
Computing h
We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative
We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.
The minimum area is