area of a parking lot: One campus of Houston community college has plans to construct a rectangular parking lot on land bordered on one side by a highway. there are 640 feet of fencing available to fence the sides. let x represent the length of each off the two parallel sides of fencing.
a)Draw the illustration of the problem
b) express the length of the remaining side to be fenced in terms of x.
c)what are the restrictions of x?
d)determine the values of x that will give an area between 30,000 and 40,000 feet squared.
e)what dimensions will give a maximum area, and what will this area be?
f)determine a function A that represents the area of the parking lot in terms of x?
Just take it step by step.
total fence: 640
two parallel sides: 2x
b) remaining side: 640-2x
c) obviously 640-2x>0 ==> x < 320
d) 30000 < x(640-2x) < 40000
56 < x < 85 approx
e) max area approx 51000 at x=160
f) a = x(640-2x)
thanks! i have one more that i posted about the path of a frogs leap. I can't figure it out
a) To start, draw a rectangle representing the parking lot. Label the length of each of the parallel sides as 'x' and the remaining side as 'y'.
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x
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y
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b) The remaining side to be fenced in terms of x can be found by subtracting two times the length of x from the total available fencing:
y = 640 - 2x
c) The restrictions on x can be determined by considering the practical limits for the dimensions of the parking lot. Since both x and y represent lengths, they must be positive values. Additionally, the sum of twice x and y must be less than or equal to the total available fencing of 640 feet. Mathematically, the restrictions can be expressed as:
x > 0
y > 0
2x + y <= 640
d) To find the values of x that will give an area between 30,000 and 40,000 square feet, we need to set up an equation using the area formula for a rectangle. The area (A) is given by A = x * y. Substitute the expression for y from step b) into the equation:
A = x * (640 - 2x)
Now, we can solve for x by setting up the inequality:
30,000 ≤ x(640 - 2x) ≤ 40,000
Simplify the inequality to:
15 ≤ 320x - x^2 ≤ 20
Rearrange to obtain a quadratic equation:
x^2 - 320x + 20 ≥ 0
Solve this quadratic equation to find the values of x that satisfy the area constraints.
e) To find the dimensions that give a maximum area, we need to find the maximum value of the area formula A = x * y. This can be achieved by analyzing the graph of the area function or by using calculus techniques such as differentiation or optimization methods.
f) The function A that represents the area of the parking lot in terms of x is:
A(x) = x * (640 - 2x)