Tshabalala wants to reconstruct his farm to separate his sheep and goats. Therefore he decided to enclose a rectangular field with a fence and divide it into two smaller rectangular fields by constructing another fence parallel to one side of the field. He has 4500m of fencing. Find the dimensions of the field so that the total enclosed area is maximum.
I came up with this equation A (x)= 2250-1.5x^2 but the graph should be a parabola with a maximum point. my graph is a straight line. I need help!
You don't tell me what your x represents, but from your equation I conclude that you defined x to be each of the 3 parallel sides and y as the length of the whole field
then area = xy
= x(2250 - 1.5x)
= -1.5x^2 + 2250x
your A(x) = 2250 - 1.5x^2 is almost correct,
but it is indeed a parabola and not a straight line like you claim.
(I have no idea how you graphed it as a straight line)
It should have been
A(x) = 2250x - 1.5x^2
So the x of the vertex is -b/(2a)
= -2250/-3 = 0
= 750
A(750) = 2250(750)-1.5(750)^2 = 843750
x = 750
y = 2250 - 1.5x = 1125
The length of the whole field is 1125m and each the 3 widths = 750 m
check:
3x + 2y = 4500 m as it should be
To solve this optimization problem, we need to find the dimensions of the rectangle that maximize its enclosed area. Let's break down the problem step by step:
1. Define the variables: Let's assume the width of the rectangle is 'x' meters and the length of the rectangle is 'y' meters.
2. Set up the equations:
- The equation for the perimeter: Perimeter = 2x + 3y (since there are two shorter sides and one longer side).
- The equation for the amount of fencing used: Fencing Used = Perimeter = 2x + 3y.
We know that Tshabalala has 4500m of fencing available, so we can set up the equation:
4500 = 2x + 3y.
3. Solve for one variable in terms of the other:
Rearranging the equation, we get: y = (4500 - 2x)/3.
4. Calculate the area in terms of x:
The area,A(x), of the rectangle is given by: A(x) = x * y = x * [(4500 - 2x)/3].
5. Simplify the area equation:
Let's simplify the area equation by multiplying out the terms:
A(x) = (4500x - 2x^2)/3.
6. Differentiate the area equation:
To find the maximum, we need to find the critical points by differentiating the area equation with respect to x:
d(A(x))/dx = (4500 - 4x)/3.
7. Set the derivative equal to zero and solve for x:
Setting the derivative equal to zero, we have: (4500 - 4x)/3 = 0.
Solving for x, we get: 4500 - 4x = 0, or x = 1125.
8. Substitute the value of x into the equation for y:
Plugging x = 1125 into the equation y = (4500 - 2x)/3, we get: y = 1125.
Therefore, the dimensions of the rectangle that maximize the enclosed area are x = 1125m and y = 1125m (length = width = 1125m), resulting in a square-shaped field.