Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used for the three sides is 240 ft. Let x be the length of two sides perpendicular to the wall as shown. Write an equation of area A of the enclosure as a function of the length x of the rectangular area. Then find the value(s) of x for which the area is 5500 ft^2.
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I I
x I I x
_________________________ Existing Wall
X is the shorter wall. The existing wall is longer than the one across from it, I tried to make it look like that in the picture.
240 = 2x+L so L = (240 - 2 x)
A = x L = x (240 - 2 x) = 240 x - 2 x^2
so
5500 = 240 x - 2 x^2
2750 = 120 x - x^2
x^2 -120 x + 2760 = 0
I get about 89 or 31
solve quadratic
To find the equation of the area A of the enclosure as a function of the length x, we can start by calculating the length of the longer wall, which is equal to 240 ft minus the lengths of the two perpendicular sides, each of length x. Since we have three sides in total, the length of the longer wall is:
Length of longer wall = 240 ft - 2x ft
The area of the enclosure is given by the multiplication of the lengths of the two perpendicular sides. In this case, it would be:
Area A = x ft * (240 ft - 2x ft)
Next, we need to find the value(s) of x for which the area is 5500 ft^2. We can set up the equation:
x * (240 - 2x) = 5500
Simplifying and rearranging this equation, we have:
240x - 2x^2 = 5500
Rearranging further, we obtain a quadratic equation:
2x^2 - 240x + 5500 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring might not be straightforward in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, the variables a, b, and c correspond to the coefficients as follows:
a = 2
b = -240
c = 5500
Plugging in these values, we can solve for x by substituting them into the quadratic formula.
To find the equation of the area A as a function of the length x, we need to determine the dimensions of the rectangular enclosure.
From the given information, we know that the total length of the fence used for the three sides is 240 ft. Since two sides perpendicular to the wall have a length of x each, the remaining side (parallel to the wall) must have a length of (240 - 2x) ft.
Therefore, the dimensions of the rectangular enclosure are x ft by (240 - 2x) ft.
To find the area A, we multiply the length and width of the enclosure:
A = x * (240 - 2x)
To find the value(s) of x for which the area is 5500 ft^2, we can set the equation A = 5500 and solve for x:
x * (240 - 2x) = 5500
Expanding and rearranging the equation:
240x - 2x^2 = 5500
Rearranging to standard quadratic form:
2x^2 - 240x + 5500 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring might not be straightforward, so let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values a = 2, b = -240, and c = 5500 into the formula:
x = (-(-240) ± sqrt((-240)^2 - 4*2*5500)) / (2*2)
Simplifying:
x = (240 ± sqrt(57600 - 44000)) / 4
x = (240 ± sqrt(13600)) / 4
x = (240 ± 116.6) / 4
Now we have two possible solutions for x:
1. x = (240 + 116.6) / 4 = 89.15 ft
2. x = (240 - 116.6) / 4 = 30.85 ft
Therefore, the values of x for which the area is 5500 ft^2 are approximately 89.15 ft and 30.85 ft.