A farmer is building a rectangular garden bed next to a river and has 8 metres worth of fence line to fence it off.
I understand that the equation of this would be:
A = w(8 - 2w)
However, I am unsure what to do next
depends on what the question is, no?
oh, I'm sorry!
The question is asking to find the dimensions needed for the maximum area
A = 8w - 2w^2
As you recall, the vertex of a parabola (in this case, the maximum area) is achieved at x = -b/2a
In this case, that's x = -8/-4 = 2
Or, you can note that the roots of A=0 are 0 and 4. The vertex is midway between the roots, at x=2.
To solve the problem, you need to find the dimensions of the rectangular garden bed that maximize its area.
The formula you mentioned, A = w(8 - 2w), represents the area of the garden bed, where "w" represents the width of the bed.
To find the dimensions that maximize the area, you need to find the value of "w" that maximizes the expression A = w(8 - 2w).
To do this, you can apply a method called calculus, specifically the derivative, to find the critical points of the function A(w). The critical points are where the derivative is equal to zero or undefined.
1. Take the derivative of A with respect to w:
dA/dw = 8 - 4w
2. Set the derivative equal to zero and solve for "w":
8 - 4w = 0
4w = 8
w = 8/4
w = 2
So, the width of the rectangular garden bed that maximizes the area is 2 meters.
To find the length, you can substitute the value of "w" back into the original equation:
A = w(8 - 2w)
A = 2(8 - 2(2))
A = 2(8 - 4)
A = 2(4)
A = 8
Therefore, the length of the rectangular garden bed is 8 meters.
In summary, the dimensions of the rectangular garden bed that maximize its area are 2 meters for the width and 8 meters for the length.