Respuesta :
Answer:
a). (680 - 2x) ft
b). 0 < x < 340
c). A(x) = x(680 - 2x)
d). 32.52 < x < 107.55 or 232.45 < x < 302.48
e). 170 ft, 170 ft, 340 ft.
Step-by-step explanation:
(a). Remaining side of the triangle will be = (680 - 2x) ft
(b). Since 680 - 2x > 0
2x < 680
x < 340
and x > 0
So 0< x < 340 will be the restriction for x
(c) Let the area of the parking lot is given by A(x) then
A(x) = x(680 - 2x) will be the area of the rectangular park.
(d) If area is between 20000 and 50000 square feet then
20000 < x(680 - 2x) < 50000
and we have to determine the values of x.
Since 20000 < 680x - 2x² < 50000
10000 < 340x - x² < 25000 [by deviding the inequality by 2]
Now we can break this inequality t solve for the value of x.
10000 < 340x - x² and 340x - x² > 25000
-x² + 340x > 10000
or x² - 340x + 10000 < 0
Now we convert this inequality into a whole square form
x² - 2(170x) + 10000 + 18900 < 18900
(x - 170)² < 18900
170 - √18900 < x < 170 + √18900
32.52 < x < 302.48 -----------(1)
and -x² + 340x < 25000
x² - 340x + 25000 > 0
(x - 170)² > 3900
So (x - 170) < -(√3900) or (x - 170) > √3900
x < 107.55 or x > 232.45 -------(2)
From inequalities 1 and 2
32.52 < x < 107.55 or 232.45 < x < 302.48
(e) Since area of the rectangular parking lot is A(x) = 680x - 2x²
To calculate the maximum area of the parking we will find the derivative of the area and equate it to zero.
then [tex]\frac{dA}{dx}=\frac{d(680x-2x^{2})}{dx}[/tex] = 0
680 - 4x = 0
4x = 680
x = 170
So dimensions of the parking will be 170 ft, 170 ft, 340 ft.
