SOLUTION: A determined gardener has 110 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at leas

Algebra ->  Inequalities -> SOLUTION: A determined gardener has 110 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at leas      Log On


   

Question 992341: A determined gardener has 110 ft of deer-resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area that is enclosed to be at least 700 ft2. What range of values (in ft) is possible for the length of her garden? (Enter your answer using interval notation.)
Found 2 solutions by josgarithmetic, solver91311:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
w width, L length
system%282w%2B2L=110%2CwL%3E=700%29

L=55-w OR w=55-L, because initially w and L are not necessarily distinct.

wL%3E=700
%2855-L%29L%3E=700
-L%5E2%2B55L-700%3E=0
highlight_green%28L%5E2-55L%2B700%3C=0%29

The boundary roots for the inequality:
%2855%2B-+sqrt%2855%5E2-4%2A700%29%29%2F2
%2855%2B-+15%29%2F2
20 or 35

L is used in what becomes a quadratic inequality and in the relation formed, we must have the values BETWEEN the roots, INCLUSIVELY. Both roots are positive, so there is no other restriction to manage:
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Length of the garden is between 20 and 35 feet, inclusive.
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Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Using the formula for the perimeter of a rectangle, it can be easily shown that the sum of the length and the width of a rectangle is equal to one-half of the perimeter, which is to say:



Or, in the case of this problem:



Since the area is the length times the width,



gives the area of any rectangle that can be formed with a perimeter of 110 as a function of the width of the rectangle.

We seek the width, from which we can determine the length, of a rectangle with a perimeter of 110 feet, such that the area is greater than or equal to 700 square feet. Hence:



Find the zeros of the corresponding quadratic equation:



Since the area function given above is a quadratic function with a negative lead coefficient, the graph is a parabola that opens downward. Hence, the portion of the graph that lies between the two zeros of the function, inclusive is the portion of the quadratic inequality that is greater than zero. So the zeros are the limits of the closed interval where the area is 700 square feet or greater.

.

John

My calculator said it, I believe it, that settles it