Show all work
You have 92 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side alongthe river, find the length and width of the plot that will maximize the area.
If length along river is y, and width =x,
2x+y = 92
a = xy = x(92-2x)
This is a parabola with roots 0,46.
So, maximum value is reached at x=23.
So, the pen is 23x46
sam can paddle a canoe 30km upstream in 5 hours. the return ship takes 3 hours. find the speed of canoe in still water
Well, I must say, this is quite a puzzling situation - fencing a plot next to a river without fencing the side along the river. It's almost like inviting the river to come in for a dip! But fear not, I shall help you with this mathematical conundrum.
Let's denote the length of the plot by L and the width by W. Since we are not fencing the side along the river, we will have two sides of length L and one side of length W. The total fencing length is given as 92 feet, so we can write the equation:
2L + W = 92
Now, in order to maximize the area, we need to find an expression for the area of the plot. The area of a rectangle is given by the formula A = L * W. Since we want to maximize the area, we need to find an expression for A in terms of a single variable.
Using the equation above, we can express W in terms of L:
W = 92 - 2L
Substituting this expression for W in the formula for the area, we get:
A = L * (92 - 2L)
To find the maximum area, we need to find the value of L that maximizes this expression. We can do this by finding the vertex of the parabola.
To do that, we can rewrite the equation in standard form:
A = -2L^2 + 92L
Now, the vertex of a parabola in the form y = ax^2 + bx + c can be found using the formula x = -b / (2a). In our case, a = -2 and b = 92, so plugging these values into the formula, we get:
L = -92 / (2 * -2)
Simplifying this expression, we find that the value of L that maximizes the area is L = 23.
Now, we can substitute this value of L back into the equation for W to find the corresponding width:
W = 92 - 2 * 23 = 92 - 46 = 46
So, the length of the plot that will maximize the area is 23 feet, and the width is 46 feet.
But remember, my dear friend, to be cautious about letting the river in without any protection. You don't want your plot to turn into a makeshift water park!
To maximize the area of the plot, we can use the fact that the perimeter of a rectangle is given by the formula:
Perimeter = 2 * (length + width)
In this case, we are given that the total length of the fence is 92 feet. Since one side of the plot borders the river and does not need to be fenced, we can write the equation:
Perimeter = length + 2 * width
Substituting in the given perimeter of 92 feet:
92 = length + 2 * width
Solving this equation for length, we get:
length = 92 - 2 * width
Now we can express the area of the plot in terms of length and width. The area of a rectangle is given by the formula:
Area = length * width
Substituting the expression for length from above:
Area = (92 - 2 * width) * width
Expanding this equation, we get:
Area = 92 * width - 2 * width^2
To maximize the area, we can take the derivative of the area equation with respect to width and set it equal to zero.
d(Area)/d(width) = 92 - 4 * width
Setting this equation equal to zero, we have:
92 - 4 * width = 0
Solving for width, we find:
4 * width = 92
width = 92 / 4
width = 23
Substituting this value of width into the equation for length, we get:
length = 92 - 2 * 23
length = 92 - 46
length = 46
Therefore, in order to maximize the area, the plot should have dimensions of 46 feet by 23 feet.
To solve this problem, we need to use optimization techniques. Let's see how we can proceed step by step:
1. Define the problem:
We want to find the dimensions of a rectangular plot that will maximize the area, given that we have 92 feet of fencing to enclose the other three sides of the plot (not including the side along the river).
2. Determine what needs to be optimized:
In this case, we want to maximize the area of the plot. The area of a rectangle is given by the formula: A = length x width.
3. Identify the variables:
Let's assume the length of the plot is represented by 'L' and the width is represented by 'W'.
4. Set up the constraint equation:
We know that the total amount of fencing available is 92 feet. This means that the sum of the three sides (not including the river) should equal 92 feet:
2L + W = 92
5. Express the objective function in terms of a single variable:
Since we want to optimize the area, we can express the area of the plot in terms of a single variable using the constraint equation. Solve the constraint equation for W:
W = 92 - 2L
Now, substitute the expression for W into the area formula:
A = L x W = L x (92 - 2L) = 92L - 2L^2
6. Find the derivative of the objective function:
To find the maximum value of A, we need to find the critical points. We can do this by finding the derivative of A with respect to L and setting it equal to zero:
dA/dL = 92 - 4L = 0
7. Solve for L:
Solving the equation 92 - 4L = 0, we get:
4L = 92
L = 23
8. Find the corresponding width:
Substitute the value of L into the constraint equation to find the value of W:
2L + W = 92
2(23) + W = 92
W = 92 - 46
W = 46
9. Verify the solution:
To ensure that the obtained values for L and W maximize the area, you can compute the area using these values and compare it to the area from any neighboring points. In this case, we don't have any neighboring points within the given constraints, so there is no need for further verification.
Therefore, the length of the plot that will maximize the area is 23 feet, and the width is 46 feet.