A rectangular garden with an area of 250 square meters is to be located next to a building and fenced on three sides, with the building acting as a fence on the fourth side. Let x be the length of the side parallel to the building.
1. Find T(x), the total amount of the fence needed to enclose the garden.
2. If the total amount of fencing needed is 60 meters, determine all possible dimensions of the garden.
I need help in figuring out exactly how to go about solving this problem because I don't know where to start.
Area= xW=250
x= 250/W
T= 2W + x
T=2W+250/w =60
solve for w, notice it will be a quadratic, and have two solutions.
2I-6U-A<A solve it for a message
X=5 , X+25
To solve this problem, we need to use the information given and use some algebraic equations to find the answers.
1. To find T(x), the total amount of fence needed to enclose the garden, we need to consider the three sides that need to be fenced. The fourth side is the building, so we don't need to include that in our calculation.
Let's denote the length of the side of the garden perpendicular to the building as y. Given that the area of the garden is 250 square meters, we have xy = 250.
Since the garden is fenced on three sides (perimeter) excluding the side adjacent to the building, the total amount of fence needed is equal to the sum of the three sides: T(x) = x + y + x = 2x + y.
2. If the total amount of fencing needed is 60 meters, we can set up an equation using T(x) and solve for the dimensions of the garden.
Based on what we derived in step 1, we have 2x + y = 60.
Now we can substitute y = 250/x into the equation to obtain 2x + 250/x = 60.
Multiply both sides of the equation by x to eliminate the fraction: 2x^2 + 250 = 60x.
Rearrange the equation to form a quadratic equation: 2x^2 - 60x + 250 = 0.
Factorizing the quadratic equation, we get (2x - 10)(x - 25) = 0.
So, x = 10 or x = 25.
Substituting these values back into y = 250/x, we get y = 250/10 = 25 or y = 250/25 = 10.
Therefore, the possible dimensions of the garden are:
- x = 10 meters, y = 25 meters
- x = 25 meters, y = 10 meters.
These are the solutions to the problem.