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Identify the quantity to be maximized or minimized. Write an expression for this quantity. Maximize the area. A = xy
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Optimization ProblemsExample 1:
A rancher has 300 yards of fencing material and wants to use it to enclose a rectangular region. Suppose the above region is bordered by a river so that fencing is only needed on three sides. What dimensions would give a region of maximum area?
Solution
Let x be the width of the region and y be the length. Let the perimeter be P and the area A.
x
y
x
Draw a diagram and introduce variables for the quantities involved.
Identify the quantity to be maximized or minimized. Write an expression for this quantity.
Maximize the area. A = xy
Identify constraints and write an expression for the constraints.
The constraints are that the perimeter can only be 300 yards. (that is all the fencing that we have)
Therefore: P = 2x +y
300 = 2x + y (since the perimeter is 300)
Rewrite the Constraint formula• 300 = 2x + y• y = 300 –2x• Also x and y must be greater than zero.• x > 0 , y >0• Since y = 300 –2x, and y > 0• 300 –2x > 0• 300 > 2x• x < 150• So 0 < x < 150• These are the endpoints that we must check later.
Write the expression to be maximized as a FUNCTION.
• Use the constraint formula to substitute into the maximized expression.
• A = x y• Substitute y = 300 – 2x into A = xy• A = x (300 –2x)
Find the critical points for this function.
• A(x) = x (300 –2x)• A(x) = 300x – 2x2
• A’(x) = 300 – 4x• Set A’(x) = 0 to find critical values.• 300 – 4x = 0• 300 = 4x• x = 75• A(75) = 300(75)-2(75)2= 11250• The critical point is (75,11250)
Graph of A(x)=300x-2x2
Determine if this is a maximum or a minimum.
• Use the second derivative test• A’’(x) = -4• Since A’’(x) < 0 for all x values, (75,
11250) is a maximum.
Check the Endpoints
• A(0)= 0• A(150) = 0• A(75) = 11250• Therefore when x = 75 there is an absolute
maximum for the given interval.
State the Solution.• Substitute x = 75 into y = 300 –2x• y = 300 – 2(75)• y = 150• Therefore the dimensions that will
maximize the area of the region is a width of 75 yards and a length of 150 yards.