SOLUTION: You plan to put a fence around a rectangular lot. The length of the lot must be at least 70ft. The cost for the fence along the length of the lot is $2 per foot, and the cost of th

You plan to put a fence around a rectangular lot. The length of the lot must be
at least 70ft. The cost for the fence along the length of the lot is $2 per
foot, and the cost of the fence along the width is $3 per foot. The total cost
can not exceed $360. 
a. Use two variables to write a system of inequalities that models the problem

Let x = the length and y = the width
      
 ___________
|     x     | 
|           |y
|           |
|___________|

>>...The length of the lot must be at least 70ft...<<

Translation:  x >= 70

>>...The cost for the fence along the length of the lot is $2 per foot,...<<

Translation: cost along the two lengths is $2 times 2x (there are two
lengths). This amounts to 2(2x) or 4x.

>>...the cost of the fence along the width is $3 per foot...<<

Translation: cost along the two widths is $3 times 2y (there are two
widths). This amounts to 3(2y) or 6y.

>>...The total cost can not exceed $360...<<

Translation 4x + 6y <= 360

We must also require that the variables x and y are not negative. x >= 70
meets this requirement, so we only need y >= 0

x >= 70
4x + 6y <= 360 
y >= 0

These are the inequalities which model the problem.

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b. Graph the system, and shade the feasible region.

Draw the graphs of the boundary lines

x = 70           a vertical line through (70,0)
4x + 6y = 360    a slanted line from (0,60) to (90,0)
y = 0            the x-axis

The feasible region is a triangle whose corners are

(70, 0), (70, 13 1/3) and (90,0)

Shade this triangle. 

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c.What is the maximum width of the lot if the length is 60 feet?

y is the width and the maximum value must occur at one of the corner
points.  Thus the maximum value of y is at the corner point (70, 13 1/3),
so the maximum width is 13 1/3 feet.  (A very narrow lot, really a strip)

Edwin
AnlytcPhil@aol.com