1. 2.7 Mathematical Models

2. Optimization Problems 1)Solve the constraint for one of the variables 2)Substitute for the variable in the objective Function Optimization Problem: Constraints: S.T.:

3. Recall: Box Project An open box with a square base is to be cut from a 8.5 inch by 5.5 inch paper 1) Write the Optimization problem to maximize the volume length x width  8.5 inches   5.5 inches 

4. Modeling Area p. 135 #8 A farmer has 3000 feet of fencing to enclose a rectangular field. One side lies along a river, so only three sides need fencing.

5. Modeling Area A farmer has 3000 feet of fencing to enclose a rectangular field. One side lies along a river, so only three sides need fencing. 2) For what value of l is the area the largest? x 1) Express the area A of the field enclosed by the fencing as a function of l, the length of the side parallel to the river. w

6. 2.7 (continued) Modeling Distance Let P = (x,y) be a point on the graph of y = 2/x 1) Express the distance d from P to the origin as a function of x. 2) For what value(s) of x is d the smallest?

7. Modeling Distance Let P = (x,y) be a point on the graph of y = 2/x 2) For what value(s) of x is d the smallest?

8. Modeling Area A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be the point in Quadrant I that is a vertex of the rectangle and is on the circle. 1) Express the area A of the rectangle as a function of x. 2) For what value of x is A the largest? What about P ?

9. Modeling Area A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be the point in Quadrant I that is a vertex of the rectangle and is on the circle. 2) Express the perimeter P of the rectangle as a function of x. 2x2x

10. Modeling Area A rectangle is inscribed in a semicircle of radius 2. Let P = (x,y) be the point in Quadrant I that is a vertex of the rectangle and is on the circle. 3) For what value of x is A the largest? What about P ? 2x2x

11. Revenue Revenue = (price) x (number of items sold) p. 135 #6