Question 40752
Your answer to part a) is correct, but not part b) I'm afraid.
Let x = vertical dimension
Let y = horizontal dimension
a) total length of fencing is 600 = 3x + y
.: y = 600 - 3x
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b) Enclosed area, A = length times width
A = xy
A = x(600-3x) , x <= 200
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c) you are given a value for the area required. So set A = 22,500
Now, A = x(600-3x)
So,
22,500 = x(600-3x)
22500 = 600x - 3x²
x² - 200x + 7500 = 0
(x-50)(x-150) = 0
x = 50, x = 150
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using y = 600-3x,
y = 450, y = 150
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So there are two solutions. Both solutions are valid. They both give an enclosed area of 22,500 square feet
d) A = 600x - 3x²
Now differentiate the expression for A and set to zero to get a turning point.
dA/dx = 600 - 6x
setting dA/dx = 0 gives,
600 - 6x = 0
x = 100
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Check to see that this is a maximum, by differentiating again. If d²A/dx² is negative at x = 100, then the turning point is a maximum. (If d²A/dx² is positive at x = 100, then the turning point is a minimum)
dA/dx = 600 - 6x
d²A/dx² = -6, which is < 0 therefore a maximum.
Since x = 100, y = 600 - 3x = 600 - 300 = 300
y = 300
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area = xy
Max area = 90,000 square feet
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