Arectangular area is to be enclosed and divided into thirds. the family has $760 to spend for the fencing material. the outside fence costs $10 per running foot installed, and the dividers cost $20 per running foot installed. what are the dimensions that will maximize the area enclosed?

Step-by-step explanation:

Derivatives and Optimization  

We can find where a function f has points of maxima/minima by using the first derivative criteria, i.e. f'=0 and evaluate its critical points.

Let's say the rectangular area has dimensions x (height) and y (width) and we are going to split the width into thirds. The area of that rectangle is

The perimeter of that rectangle is the length of the outside fence

It costs $10 per feet, so the cost of the outside fence is

The dividers cost $20 per foot, and we have assumed we split the width, so each divider has a length of x feet. The cost of the internal dividers is

The total cost of the fencing is

We know there is a limit of $760 to spend for the fencing material, so

Reducing

Solving for y

The area can be now expressed in terms of x alone

To find the critical point, and possible maxima/minima, we set A'=0

We find:

Since

The second derivative is

A''=-6

This means the area is maximum at the critical point

The dimensions that maximize the area are:

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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