SOLUTION: Lee has a rectangular plot to use for a flower garden. She plans to include a gravel border of uniform width around the flower garden so visitors can walk completely around the f

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Question 564768: Lee has a rectangular plot to use for a flower garden. She plans to include a gravel
border of uniform width around the flower garden so visitors can walk completely around
the flowers to view them more easily. The total size of the garden and gravel border is to
be 15 feet by 21 feet. If Lee has enough gravel to cover 117 square feet for the border,
how wide will the gravel border be? What will the dimensions of the actual flower garden
be? Round answers to nearest tenth.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
let x = the width of the border.
the total area including the border is equal to 315 square feet.
the area of the border is equal to 117 square feet.
subtract 117 from 315 and you get 198 square feet for the garden.
the length of the garden will be equal to 21 - 2x.
the width of the garden will be equal to 15 - 2x.
since the area of the garden is equal to the length times the width, then the equation for the area of the garden will be:
(21 - 2x) * (15 - 2x) = 198 square feet.
we multiply these factors out to get:
4x^2 - 72x + 315 = 198
we subtract 198 from both sides of this equation to get:
4x^2 - 72x + 117 = 0
this doesn't factor well so we use the quadratic formula to get:
x = 16.19374728 or x = 1.806252715
x = 16.19374728 doesn't work because 21 - 2*16 is negative.
the answer has to be x = 1.806252715 or there is no solution.
we go back to our equation for the area of the garden.
it is:
(21 - 2x) * (15 - 2x) = 198 square feet.
substituting 1.806252715 for x gets us:
21 - 2*1.806252715) * (15 - 2*1.806252715) = 198
simplifying this equation gets us:
17.38749457 * 11.38749457 = 198
simplifying this further gets:
198 = 198.
this confirms the value for x is good.
we confirm even further by determining the area of the border.
that will be:
4 * x^2 plus 2 *(21 - 2x) * x plus 2 * (15-2x) * x
substituting for x in this equation gets us 117 square feet.
the area of the garden is 198 square feet.
the area of the border is 117 square feet.
the value of x = 1.806252715 feet.
x is the width of the border.
the length of the garden will be 21 - 2x = = 17.38749457
the width of the garden will be 15 - 2x = 11.38749457
the area of the garden is equal to (21-2x) * (15-2x) = 198 square feet.
the equation we solved using the quadratic formula is:
4x^2 - 72x + 117 = 0
in that equation:
a = 4
b = -72
c = 117
the quadratic formula is:
x =
your answer need to be to the nearest tenth of a foot.
that makes your answer:
the width of the border is equal to 1.8 feet.
the length of the garden is equal to 17.4 feet
the width of the garden is equal to 11.4 feet.