A rectangular field is to be fenced in on four sides with a fifth piece of fencing placed Parallel to one of the shorter sides, so that the field is split in two parts. If 1600 m of fencing is available, find the largest possible area for this enclosure. What dimensions give this maximum area?
let the length be y m
and each of the shorter pieces (the width) be x
we know 2y + 3x = 1600
y = 800 - 3x/2
area = xy
= x(800-3x/2)
= 800x - 3x^2 /2
find the first derivative, set it equal to zero and find x
then find y, and the area xy
xmax is on the axis of symmetry of the parabola
-b / 2a = -800 / (2 * -3/2) = 800 / 3
To find the largest possible area for the enclosure, we need to determine the dimensions of the rectangular field that maximize the area. Let's break down the problem into steps:
Step 1: Assign variables
Let's assume the width of the rectangular field is 'x' meters.
Then, the length of the rectangular field will be 'y' meters.
Step 2: Calculate the total amount of fencing used
The total amount of fencing used would be the sum of the four sides of the rectangular field plus the fifth piece of fencing parallel to one of the shorter sides:
2x + y + y + x = 2x + 2y = 1600 m
Step 3: Simplify the equation
We can simplify the equation further by dividing both sides by 2:
x + y = 800 m
Step 4: Express one variable in terms of the other
From the simplified equation, we can rearrange it to express one variable in terms of the other. Let's express 'y' in terms of 'x':
y = 800 - x
Step 5: Calculate the area of the enclosure
The area of the rectangular field can be calculated by multiplying the length and width:
Area = width × length
Area = x × (800 - x)
Area = 800x - x^2
Step 6: Determine the maximum area
To find the maximum area, we need to find the maximum value of the area function. The maximum value will occur when the derivative of the area function with respect to 'x' equals zero.
Differentiating the area function:
d(Area)/dx = 800 - 2x
Setting the derivative equal to zero:
800 - 2x = 0
2x = 800
x = 400
Step 7: Calculate the corresponding value of 'y'
Using the simplified equation from step 3, we can calculate the corresponding value of 'y':
y = 800 - x
y = 800 - 400
y = 400
Step 8: Calculate the maximum area
Now, substitute the values of 'x' and 'y' into the area function:
Area = 800x - x^2
Area = 800(400) - (400)^2
Area = 320,000 - 160,000
Area = 160,000 m^2
Therefore, the largest possible area for this enclosure is 160,000 square meters. The dimensions that give this maximum area are a width of 400 meters and a length of 400 meters.