A security fence encloses a rectangular are on one side of a park in a city. Three sides of fencing are used, since the fourth side of the area is formed by a building. The enclosed area measures 450 square feet. Exactly 60 feet of fencing is used to fence in three sides of this rectangle. What are the possible dimensions that could have been used to construct this area?
well, 450 = 30*15, and 15+30+15=60
To solve this problem, we can use algebra to represent the dimensions of the rectangular area. Let's assume the length (L) of the rectangle is the side parallel to the building and the width (W) is the side perpendicular to the building.
We know that the enclosed area measures 450 square feet, so we have the equation:
L * W = 450 ...........(Equation 1)
We also know that exactly 60 feet of fencing is used to fence in three sides of the rectangle. The three sides are the two sides parallel to the building (L) and one side perpendicular to the building (W). Since the fourth side is formed by the building, it does not require fencing. Therefore, we have the equation:
2L + W = 60 ...........(Equation 2)
Now we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the possible dimensions.
We can start by solving Equation 2 for W:
W = 60 - 2L
Next, substitute this value of W into Equation 1:
L * (60 - 2L) = 450
Distribute the L:
60L - 2L^2 = 450
Rearrange the equation and set it equal to 0:
2L^2 - 60L + 450 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is the simplest method. Divide every term by 2 to simplify the equation:
L^2 - 30L + 225 = 0
Factor the equation:
(L - 15)(L - 15) = 0
This gives us a repeated factor of (L - 15) = 0. This means L = 15.
Now, substitute this value of L into Equation 2 to find W:
2(15) + W = 60
30 + W = 60
W = 30
Therefore, the possible dimensions for the rectangular area are L = 15 feet and W = 30 feet.
To double-check, you can substitute these dimensions into Equation 1 to ensure that the area is indeed 450 square feet:
15 * 30 = 450
The area does equal 450 square feet, confirming that the dimensions are correct.