LarCalc10 Ch03 Sec7
14-Oct-14
Applications of Differentiation
Optimization Problems
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Objective
Solve applied minimum and maximum problems.
Applied Minimum and Maximum
Problems
3
4
1
14-Oct-14
Applied Minimum and Maximum Problems
Example 1 – Finding Maximum Volume
One of the most common applications of calculus involves the determination of minimum and maximum values.
A manufacturer wants to design an open box having a square base and a surface area of 108 square inches, as shown in Figure 3.53. What dimensions will produce a box with maximum volume?
5
Example 1 – Solution
Example 1 – Solution
Because the box has a square base, its volume is
V = x2h.
Figure 3.53
6
cont’d
Because V is to be maximized, you want to write V as a function of just one variable.
Primary equation
To do this, you can solve the equation x2 + 4xh = 108 for h in terms of x to obtain h = (108 – x2)/(4x).
This equation is called the primary equation because it gives a formula for the quantity to be optimized.
Substituting into the primary equation produces
The surface area of the box is
S = (area of base) + (area of four sides)
S = x2 + 4xh = 108.
Secondary equation
7
8
2
14-Oct-14
Example 1 – Solution
cont’d
Before finding which x-value will yield a maximum value of
V, you should determine the feasible domain.
Example 1 – Solution
cont’d
To maximize V, find the critical numbers of the volume function on the interval
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 critical numbers are x = ±6.
So, the feasible domain is
You do not need to consider x = –6 because it is outside the domain. 9
Example 1 – Solution
cont’d
10
Applied Minimum and Maximum Problems
Evaluating V at the critical number x = 6 and at the endpoints of the domain
Applications of Differentiation
Optimization Problems
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Objective
Solve applied minimum and maximum problems.
Applied Minimum and Maximum
Problems
3
4
1
14-Oct-14
Applied Minimum and Maximum Problems
Example 1 – Finding Maximum Volume
One of the most common applications of calculus involves the determination of minimum and maximum values.
A manufacturer wants to design an open box having a square base and a surface area of 108 square inches, as shown in Figure 3.53. What dimensions will produce a box with maximum volume?
5
Example 1 – Solution
Example 1 – Solution
Because the box has a square base, its volume is
V = x2h.
Figure 3.53
6
cont’d
Because V is to be maximized, you want to write V as a function of just one variable.
Primary equation
To do this, you can solve the equation x2 + 4xh = 108 for h in terms of x to obtain h = (108 – x2)/(4x).
This equation is called the primary equation because it gives a formula for the quantity to be optimized.
Substituting into the primary equation produces
The surface area of the box is
S = (area of base) + (area of four sides)
S = x2 + 4xh = 108.
Secondary equation
7
8
2
14-Oct-14
Example 1 – Solution
cont’d
Before finding which x-value will yield a maximum value of
V, you should determine the feasible domain.
Example 1 – Solution
cont’d
To maximize V, find the critical numbers of the volume function on the interval
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 critical numbers are x = ±6.
So, the feasible domain is
You do not need to consider x = –6 because it is outside the domain. 9
Example 1 – Solution
cont’d
10
Applied Minimum and Maximum Problems
Evaluating V at the critical number x = 6 and at the endpoints of the domain