min and max problems

Slide 1: 

Applied Minimum and Maximum Problems Finding the Maximum Volume: A manufacturer wants to design an open box having a square base and a surface area of 108 sq. in. as shown below. What dimensions would produce a box with maximum volume? We know the equation for the volume to be: V = l  w  h But, since the base of the box is square, the volume equation would be: V = x  x  h or V = x2h This equation is called the primary equation because it gives a formula for the quantity to be optimized. The surface area of the box is: S = (area of base) + (area of four sides) S = x2 + 4xh V = x2h S = x2 + 4xh primary equation Because V is to be maximized, you want to write V as a function of just one variable. To do this, you can solve the equation x2 +4xh = 108 for h in terms of x to obtain h = (108 – x2)/(4x). Substituting into the primary equation produces: V = x2h  x2((108 – x2)/(4x))  27x - x3/4

Slide 2: 

Before finding which x-value will yield a maximum value of V, you should determine the feasible domain. That is, what values of x make sense in this problem? You know that V > 0. You also know that x must be nonnegative and that the area of the base ( A = x2) is at most 108. So, the feasible domain is: To maximize V, find the critical numbers of the volume function.   You do not need to consider x = -6 because it is outside the domain. Evaluating V at the critical number 6 and at the endpoints of the domain produces: V(0) = 0, V(6) = 108, V( ) = 0. So, V is maximized when x = 6 and the dimensions of the box are 6 X 6 X 3.

Slide 3: 

In this problem you should realize that there are infinitely many open boxes having 108 square inches of surface area. To begin solving the problem, you might ask yourself which basic shape would seem to yield a maximum volume. Should the box be tall, squat, or nearly cubical? Let’s try a few:

Slide 4: 

Guidelines for Solving Applied Minimum and Maximum Problems Identify all given quantities and quantities to be determined. If possible, make a sketch. Write a primary equation for the quantity that is to be maximized or minimized. Reduce the primary equation to one having a single independent variable. This may involve the use of secondary equations relating the independent variables of the primary equation. Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. Determine the desired maximum or minimum value by the calculus techniques discussed throughout the chapter

Slide 5: 

Which points on the graph of y = 4 – x2 are closest to the point (0, 2)? The figure shows that there are two points at a minimum distance from the point (0, 2). The distance between the point (0, 2) and the point (x, y) on the graph of y = 4 – x2 is given by: Using the secondary equation y = 4 – x2, you can rewrite the primary equation as: