A farmer has 1000 feet of fence a rectangular plot of land. The plot lies along a river so that only has three sides need to be fenced. Find the largest area that can be fenced.
What's the best way to solve this?
The best way to do it is using calculus, but you are in college algebra.
You are looking for length + width + width = 1000 feet because you don't need any for the river side.
L + 2w = 1000 or L = (1000 - 2w)
A = lw
A = (1000-2w)(w)
A = 1000w - 4w^2
120000
To find the largest area that can be fenced with 1000 feet of fence, we can use the calculus concept of optimization. Let's denote the length of the plot as L and the width as W.
To solve this problem, we need to set up an equation based on the given information:
Perimeter of the plot = 1000 feet
2L + W = 1000
To find the largest possible area, we need to maximize the function that represents the area of the rectangle. The area, A, of the rectangle is given by:
A = L * W
Since we have already expressed one variable, W, in terms of the other variable, L, we can substitute this value into the area equation:
A = L * (1000 - 2L)
Now, we have an expression for the area in terms of a single variable, L. To find the maximum area, we need to find the value of L that maximizes this function. We can achieve this by taking the derivative of the area function with respect to L and setting it to zero:
dA/dL = 1000 - 4L
Setting the derivative equal to zero:
1000 - 4L = 0
Simplifying the equation:
4L = 1000
L = 250
Substituting this value back into the equation for W:
W = 1000 - 2(250) = 500
Therefore, the dimensions of the rectangle that maximize the area are L = 250 feet and W = 500 feet. To find the maximum area, substitute these values into the area equation:
A = (250) * (500) = 125,000 square feet.
Hence, the largest area that can be fenced is 125,000 square feet.
To find the largest area that can be fenced, you need to determine the dimensions of the rectangular plot that would maximize the area. Here's a step-by-step approach to solve this problem:
1. Understand the problem: The farmer has 1000 feet of fence, and only three sides of the rectangular plot (the two widths and one length) need to be fenced.
2. Define the variables: Let's assume the width of the plot is "w" feet, and the length is "l" feet.
3. Formulate the equation: We know that the total amount of fence used will be the sum of the three sides: w + w + l = 2w + l. Since the farmer has 1000 feet of fence, we can set up the equation 2w + l = 1000.
4. Solve for one variable in terms of the other: We need to express one variable (either "w" or "l") in terms of the other so that we can find the maximum area. Solving the equation from step 3 for "l" gives l = 1000 - 2w.
5. Find the area: The area of a rectangle is given by the formula A = l * w. Substituting the expression for "l" from step 4 into this formula, we have A = (1000 - 2w) * w.
6. Maximize the area: To find the maximum area, we need to maximize the equation above. You can do this by finding the derivative of A with respect to "w" and setting it equal to zero. Then, solve for "w".
7. Once you find the value of "w", substitute it back into the equation 2w + l = 1000 to find the corresponding value of "l".
8. Calculate the area: Finally, substitute the values of "w" and "l" into the formula for the area A = l * w to find the largest area that can be fenced.
By following these steps, you should be able to find the largest area that can be fenced with the given amount of fence.