Question 973099: A problem on my math homework really gave me a hard time trying to understand it because it didnt make any sense to me... this is the question..
"Suppose you want to enclose a garden with 26 m of fencing. If the back of the garden is enclosed by a stone wall, 20 m long, what size rectangle would maximize the area of the garden?"
i tried to draw a picture, but it just doesnt make any sense...unless i am making a mistake with my diagram, at first i thought it could be a "20x3" rectangle but im not sure how i did that, could you please explain this to me.. thanks!
Found 2 solutions by josgarithmetic, stanbon:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! Draw a rectangle, and make one dimension to be 20 units long, and the other dimension is made as unknown.
As given, 26 units (or meters) of fencing. AND one of the sides of the garden is already taking 20 meters of the stone wall, so that will be 20 meters of fencing not needed for the garden.
Using x as the unknown dimenison of the garden,
account for the amount of fencing to use,  .
Intuitive sense tells you that you do not want to use a length for x less than 3, because this will only DECREASE the garden area; you cannot increase x, because the calculation must made for for using all 20 meters of the stone wall.
Say y and x are the dimensions.
 for area;
 , and  , positive numbers only.
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substitute,
You want to find which x value is in the exact middle of the roots of the equation  , which will be the maximum value for where A is maximum.
Roots are 0 and 13.
Mid-value is at  ;
from this,  , for the side opposite from the stone wall.
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! "Suppose you want to enclose a garden with 26 m of fencing. If the back of the garden is enclosed by a stone wall, 20 m long, what size rectangle would maximize the area of the garden?"
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Draw the wall; label it 20 m
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Draw the other three sides of the rectangle.
The length is 20 m
That leaves you with 6 m of fencing for the two ends.
The width must be 3 m
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Ans: Assuming that 20m wall is the width of the garden,
the rectangle must be 20m by 3m.
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Note: That is not a maximizing problem.
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Cheers,
Stan H.
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