A farmer needs to enclose three sides of a garden with a fence (the fourth side is a cliff wall). The farmer has 47 feet of fence and wants the garden to have an area of 266 sq-feet. What should the dimensions of the garden be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)
2 w + L = 47
w L = 266 so L = 266/w
2 w + 266/w = 47
2 w^2 - 47 w + 266 = 0
(2w-19)(w-14) = 0
w = 9.5 or w =14
if the length is to be max, the width should be min so use w = 9.5
L= 47 - 19 = 28
Well, I must say, this farmer really wants to make an exclusive garden! I hope they have a great view of the cliff wall. Anyway, let's figure out the dimensions of this garden without driving everyone over the edge with suspense.
Let's call the width of the garden W and the length L. Since there are three sides that need to be fenced, we have W + W + L = 47. That's two sides of width and one side of length. So, we can simplify it to 2W + L = 47.
Now, let's talk area. The area of a rectangle is width times length, so W x L = 266. We're looking for the greatest possible length, so let's solve for W in the first equation and substitute it into the second equation.
From 2W + L = 47, we can isolate W: 2W = 47 - L, so W = (47 - L)/2.
Substituting this into the second equation, we get (47 - L)/2 x L = 266.
Now, let's simplify this equation. Multiply both sides by 2 and distribute: 47L - L^2 = 532.
Rearranging this equation to find L, we get L^2 - 47L + 532 = 0.
Solving this quadratic equation, we find that L = 19 or L = 28. Since we want the longest length possible, let's choose L = 28.
Now, we substitute this value of L into the equation W = (47 - L)/2 and calculate W: W = (47 - 28)/2 = 9.5.
So, the dimensions of the garden are approximately 9.5 feet by 28 feet.
I hope the farmer enjoys their garden and doesn't accidentally wander off the cliff wall while watering their plants!
Let's assume the width of the garden is x feet.
Since the length is as long as possible, it will be equal to 47 - 2x (since we are using three sides for the fence).
Now, we can set up an equation to find the area of the garden:
Area = length × width
266 = (47 - 2x) × x
Expanding the equation, we get:
266 = 47x - 2x^2
Rearranging the equation, we get:
2x^2 - 47x + 266 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring would be the easiest method.
The factors of 266 that add up to -47 are -19 and -14. Therefore, we can factor the equation as follows:
(2x - 19)(x - 14) = 0
Setting each factor equal to zero, we have two possible solutions:
2x - 19 = 0 or x - 14 = 0
Solving for x in each equation, we find:
2x = 19 or x = 14
If 2x = 19, then x = 19/2, which is not a feasible solution in this context. So we discard it.
If x = 14, then the width of the garden is 14 feet. The length can be calculated as 47 - 2x = 47 - 2(14) = 47 - 28 = 19 feet.
Therefore, the dimensions of the garden should be 14 feet (width) and 19 feet (length).
To find the dimensions of the garden, we can use the formula for the area of a rectangle:
Area = Length * Width
Given that the area of the garden is 266 sq-feet, let's denote the length as L and the width as W.
We know that the width of the garden requires two sides of fencing, and the length requires one side of fencing. Since the fourth side is a cliff wall and does not require any fencing, we can calculate the total length of fencing needed as follows:
Total Length of Fencing = 2 * Width + 1 * Length
According to the problem, the farmer has 47 feet of fencing available, so we can set up the following equation:
2W + L = 47
Solving this equation will give us the relationship between the length and width of the garden.
Next, we can substitute the value of L in terms of W in the area formula:
266 = L * W
Now, we have a system of two equations:
2W + L = 47
L * W = 266
To solve this system, we can first solve one equation for a variable and substitute it into the other equation.
Let's solve the first equation for L:
L = 47 - 2W
Now we can substitute this expression for L into the second equation:
(47 - 2W) * W = 266
Now we have a quadratic equation. Simplifying and rearranging, we get:
47W - 2W^2 = 266
Rearranging again, we have:
2W^2 - 47W + 266 = 0
Solving this quadratic equation will give us the values of W (the width) that satisfy the given conditions.
Using factoring, quadratic formula, or other appropriate methods, we find that the solutions are W = 7 and W = 19.
Since the problem states that the length should be as long as possible, we will choose the larger value of W, which is 19.
Substituting W = 19 back into the equation 2W + L = 47, we get:
2 * 19 + L = 47
38 + L = 47
L = 9
Therefore, the dimensions of the garden should be 9 feet by 19 feet.