A farmer has 80 feet of fencing, which she plans to use to fence in a plot of land for a pigpen. If she chooses to enclose a plot along the broad side of her barn, what is the largest area that can be enclosed? (Note: The side along the barn will not require fencing)
let the width of the field be x ft
let the length be y ft
so y + 2x = 80
or y = -2x + 80
area = xy
= x(-2x+80)
= -2x^2 + 80x
do you know Calculus?
if so ...
d(area)/dx = -4x + 80 = 0 for a max area
4x = 80
x = 20
width is 20 ft, length is 40 ft.
largest area = 20(40) = 800 ft^2
If you don't know Calculus, complete the square
Well, if the farmer wants to enclose a plot along the broad side of her barn, she just needs to put up fencing on the other three sides. Let's call the length of the plot "x" and the width "y."
So, the perimeter of the plot would be x + 2y (since one side is along the barn). But we know the farmer only has 80 feet of fencing, so x + 2y = 80.
Now, to find the largest area enclosed, we need to maximize the value of xy. But it seems like the answer is running away from us...oh wait! It's a pigpen, not a track meet!
Alright, back to business. Let's solve for one of the variables in terms of the other. We can rearrange the perimeter equation to get x = 80 - 2y.
Substitute this value of x into the area equation: A = xy = (80 - 2y)y.
Now, we just need to find the maximum value of A. Luckily, I have a PhD in Calculus of Humor.
Taking the derivative of A with respect to y gives us dA/dy = 80 - 4y.
To find the maximum, we set this derivative equal to zero: 80 - 4y = 0.
Solving for y, we get y = 20. Substituting this value back into the equation for x, we find x = 80 - 2(20) = 40.
So, the maximum area that can be enclosed is 40 * 20 = 800 square feet.
And there you have it – the farmer can have a pigpen with the largest area of 800 square feet. Just make sure the pigs don't hog all the space!
To find the largest area that can be enclosed, we need to determine the dimensions of the plot that will maximize the area.
Let's assume the width of the plot along the barn is x feet.
Since there are two equal lengths of fencing needed on either side of the barn, the remaining fencing available for the two ends of the plot is (80 - 2x) feet.
The area of a rectangle is given by the formula: Area = Length × Width.
In this case, the length is (80 - 2x) feet, and the width is x feet.
Therefore, the area of the plot is: Area = (80 - 2x) × x.
To find the largest possible area, we need to maximize this formula.
We can do this by finding the value of x that maximizes the area.
To find the value of x, we can take the derivative of the area formula with respect to x and set it equal to zero.
Let's differentiate the area formula:
d(Area)/dx = d((80 - 2x) × x)/dx
= (80 - 2x) × 1 + x × (-2)
= 80 - 2x - 2x
= 80 - 4x
Now, we set this derivative equal to zero and solve for x:
80 - 4x = 0
4x = 80
x = 20
So, the width of the plot along the barn that maximizes the area is 20 feet.
To find the maximum area, we substitute this value of x back into the area formula:
Area = (80 - 2x) × x
= (80 - 2 * 20) × 20
= (80 - 40) × 20
= 40 × 20
= 800 square feet
Therefore, the largest area that can be enclosed is 800 square feet.
To find the largest area that can be enclosed with the given 80 feet of fencing, we need to determine the dimensions of the rectangular plot.
Let's assume that the side of the plot along the barn has a length of "x" feet. Since this side doesn't require fencing, we can allocate all 80 feet of fencing to the remaining three sides.
So, the other three sides, which are parallel to each other, will require 80 feet of fencing in total. Since two sides are parallel, they will each have a length of "y" feet.
Let's break down the calculation:
- The side along the barn is x feet and doesn't require any fencing.
- The other two sides require a total of 80 feet of fencing, so each would be y feet.
- Since there are two sides of length y, the total length of these sides would be 2y.
- Therefore, 2y = 80 feet, which means y = 40 feet.
So, we have the dimensions of the plot: x for the side along the barn and 40 for the other two sides.
The area of a rectangle is given by the formula: Area = length x width.
Therefore, the total area enclosed by the fencing can be calculated as: Area = x * 40.
To find the largest possible area, we need to maximize the value of x. Since the farmer has a fixed length of fencing (80 feet), the largest area can be obtained when x is half of the total length of the fencing. In this case, x = 80/2 = 40 feet.
Thus, the largest area that can be enclosed by the fencing is: Area = 40 feet (x) * 40 feet = 1600 square feet.