A rectangular field is to be enclose by 800 meters of fencing and the fourth side with be along the side of a barn.
A) What dimensions will give the maximum area?
B) What is the maximum area?
let the single side by y
let the two other sides be x each
so 2x+y = 800
y = 800-2x
area = xy
= x(800-2x)
= -2x^2 + 800x
Since you rated this grade 10 , I will assume you don't have Calculus, so let's complete the square
area = -2(x^2 - 400x <>+40000-40000)
= -2( (x-200)^2 - 40000)
= -2(x-200)^2 + 80000
the max area is 80,000 m^2 and is obtained when
x = 200 m, which makes y = 800-2(200) = 400 m
To find the dimensions that will give the maximum area for the rectangular field, we can use the concept of calculus. The formula for the perimeter (P) of a rectangle is given by:
P = 2L + W
where L is the length and W is the width. In this problem, we are given that the perimeter is 800 meters, so we can write:
800 = 2L + W
We know that the fourth side will be along the side of a barn, so the total perimeter will be made up of three sides: two sides with length L and one side with length W.
To maximize the area (A) of the rectangle, we can use the equation:
A = L * W
Now, we can express one of the variables in terms of the other, substitute it into the area equation, and solve for the maximum area.
From the perimeter equation, we can express W in terms of L:
W = 800 - 2L
Substituting this value for W in the area equation, we get:
A = L * (800 - 2L)
Expanding the equation, we have:
A = 800L - 2L^2
To find the maximum area, we can differentiate the area equation with respect to L and set it equal to zero:
dA/dL = 800 - 4L = 0
Solving this equation, we get:
4L = 800
L = 200 meters
Now, we can find the corresponding value of W using the perimeter equation:
800 = 2(200) + W
800 = 400 + W
W = 400 meters
Therefore, the dimensions that will give the maximum area are a length (L) of 200 meters and a width (W) of 400 meters.
To find the maximum area, we can substitute these values of L and W into the area equation:
A = 200 * 400
A = 80,000 square meters
Therefore, the maximum area of the rectangular field is 80,000 square meters.