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Question 40752This question is from textbook college algebra
: I got part a and b of this question , need help with d & C
(of course a word problem) A farmer wishes to enclose a rectangular region bordering a river with fencing, as shown in the diagram. Suppose that x represents the length of each of the three parallel pieces of fencin. She has 600 ft of fencing available.
( the diagram is a rectangle, the top part is water and the rest is fencing, there is a vertical dotted line down the center, it and both the left and right side are labeled x the bottom is just fencing)
a) What is the length of the remaining piece of fencing in terms of x?
---My answer, 600-3x
b) Determine a function(A) that represents the total area of the enclosed region give any restrictions for x.
Since Length and Width cannot be negative i put 600-3x is greater than or equal to zero and solved that, 200 is greater than or equal to x, and x is greater than or equal to x so my solution is
0is less than or equal to x is less than or equal to 200.
NOW FOR THE ONES IM LOST ON>>>>>
c) What dimensions for the total enclosed region would give an area of 22,500 sqaure feet?
and
d) What is the maximum area that can be enclosed?
ELisa
This question is from textbook college algebra
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! 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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