Section 3.8 Application for finding Maxima and Minima

3.8 Application for finding Maxima and Minima

In this section, we learn to find maximums and minimums of real world problems involving in two variables. 

Example 1: Find three positive numbers the sum of which is 16, such that the sum of their squares is as small as possible.

Exercise 1: Find three positive numbers the sum of which is 27, such that the sum of their squares is as small as possible.

Example 2: Find the points on the surface $x+y-z^2=-4$ that are closest to the origin.

Exercise 2: Find the points on the surface $x-y+z^2=5$ that are closest to the origin.

Example 3: Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the plane $x+2y+3z=6.$

Exercise 3: Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on

the plane $x+y+z=1.$

Example 4: A cardboard box without a lid is to be made with a volume of $9$${\text{ft}}^3$. Find the dimensions of the box that requires the least amount of cardboard.

Exercise 4: A cardboard box without a lid is to be made with a volume of $4$${\text{ft}}^3$. Find the dimensions of the box that requires the least amount of cardboard.

Example 5: Find the point on the surface $z^2=-x^2+y^2$ that is closest to the point $(1,0,1)$.

Exercise 5: Find the points on the surface $z^2=x^2-y^2$ that is closest to the point $(0,-1,1)$.

Group work:

1. A company manufactures two types of boats: speeding boats and sailing boats. The total revenue from $x$ speeding boats and $y$ sailing boats is given by $R(x,y)=-x^2-2y^2-2xy+12x+16y$. Find the values of $x$ and $y$ to maximize the total revenue.

2. A shipping company handles rectangular boxes provided the sum of the length, width, and height of the box does not exceed $96$ in. Find the dimensions of the box that meets this condition and has the largest volume.

3. Pro- T company has developed a profit model that depends on the number $x$ of golf balls sold per month(measured in thousands), and the number of hours per month of advertising $y$, according to the function $z=f(x,y)=48x+96y-x^2-2xy-9y^2$, where $z$ is measured in thousands of dollars. The maximum number of golf balls that can be produced and sold is $50,000$, and the maximum number of hours of advertising that can be purchased is $25$. Find the values of $x$ and $y$ that maximize profit, and find the maximum profit.