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Question 1189873: 1. Richard has available 400 yards of fencing and wishes to enclose a rectangular area.
a.) Express the area A of the rectangle as a function of the width x of the rectangle?
b.) What is the domain of A?
Found 3 solutions by ikleyn, Theo, Solver92311:
Answer by ikleyn(52794)   (Show Source):
You can put this solution on YOUR website! .
Richard has available 400 yards of fencing and wishes to enclose a rectangular area.
a.) Express the area A of the rectangle as a function of the width x of the rectangle?
b.) What is the domain of A?
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(a) the Length + the Width = 400/2 = 200,
so if the length is x yards, then the width is (200-x) yards.
Then the area is A = xy square yards, which is A(x) = x*(200-x) = -x^2 + 200x.
(b) The function is A(x) = x*(200-x) = -x^2 + 200x,
and it makes sense where A(x) is non-negative and x is non-negative,
so the domain of the function A(x), defined in this way for the area, is 0 <= x <= 200.
Solved.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! a = l * w
p = 2 * (l + w)
a is the area
l is the length
w is the width
p is the perimeter
when p = 400, p = 2 * (l + w) becomes 400 = 2 * (l + w)
divide both sides of this equation by 2 to get:
200 = l + w
solve for l to get:
l = 200 - w
in the formula for area:
when l = 200 - 2, a = l * w becomes:
a = (200 - w) * w.
simplify to get:
a = 200 * w - w^2.
a must be greater than or equal to 0 (can't be negative).
if a >= 0, then:
200 * w - w^2 >= 0
add w^2 to both sides of this equation to get:
200 * w >= w^2
divide both sides of this equation by w to get:
200 >= w
if 200 >= w, then:
w <= 200
w must also be >= 0, therefore:
0 <= w <= 200
that's your domain.
the area is your range.
the domain is the width which is greater than or equal to 0 and less than or equal to 200.
that says the domain of the area is 0 <= width <= 200.
that should be your solution.
the equation of area = 200 * width minus width squared can be graph by letting y = the area and x = the width.
the formula for graphing becomes y = 200x - x^2.
the graph looks like this:
you can see from the grpah that the area, represented by y, is not negative when 0 <= x <= 200, with x representing the width.
the coordinate points on the graph are in (x,y) format.
x represents the width of the rectangle.
y represents the area of the rectangle.
at (100,10,000) the width is 100 and the area is 10,000.
you don't see the length on the graph, but you can always find the length, because length + width will always be 200.
therefore, when the width is 100, the length has to be 100.
the area is equal to length * width = 100 * 100 = 10,000, as shown on the graph.
Answer by Solver92311(821)  (Show Source):
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