A rectangular lot is bordered on one side by a stream and on the other 3 sides by 600m of fencing. The area is a maximum. Determine the area.
What are the length and width of the lot with the maximum area.
State the domain.
State the range
To determine the area of the rectangular lot with a maximum area, we can start by considering the given information.
Let's say the length of the lot is 'x' meters and the width is 'y' meters.
We know that the lot is bordered on one side by a stream, so one side of the rectangle (length) will be along the stream, and the remaining three sides will be fenced.
We are also given that the total length of the fencing is 600 meters. Since there are three sides fenced, we can set up the equation:
2x + y = 600 (equation 1)
We want to maximize the area of the rectangle, so the area 'A' is given by:
A = xy
To find the dimensions with the maximum area, we need to express one of the variables in terms of the other. From equation 1, we can solve for y:
y = 600 - 2x
Substituting this value of y into the area formula, we get:
A = x(600 - 2x)
To find the maximum area, we can take the derivative of A with respect to x and set it equal to zero:
dA/dx = 600 - 4x
Setting it equal to zero:
600 - 4x = 0
4x = 600
x = 150
Substituting this value of x back into equation 1, we can find y:
2(150) + y = 600
300 + y = 600
y = 300
Therefore, the length of the lot with the maximum area is 150 meters, and the width is 300 meters.
Now let's state the domain and range:
Domain: In this context, the domain represents the possible values of x and y. Since both x and y represent lengths, they must be positive numbers. Additionally, the sum of the two lengths (2x + y) should not exceed the total length of the fence, which is 600 meters.
Therefore, the domain is: 0 < x < 300 and 0 < y < 600.
Range: The range represents the possible values of the area 'A'. Since the area cannot be negative, the range of A is 0 < A < 150,000 square meters.
length of single side parallel to stream ----- y m
length of each of the other two sides --- x m
2x+y=600
y = 600-2x
area = xy = x(600-x) = -x^2 + 600x
At this stage, I expect you to know how to find the vertex of this downwards parabola.
Either my using Calculus, completing the square or some other method.