A farmer has 450m of fencing to enclose a rectangular area and divide it into two sections.
a) Write an equation to express the total area enclosed as a function of the width.
b) Determine the doman and range of this area function.
c) Determine the dimensions that give the maximum area.
Can someone explain how to do this please? I got part a already, and the equation I got is:
A(w)= ( 450-3w
______
2 ) w
I don't understand part b and c.
b) To determine the domain of the area function, we need to find the possible values for the width of the rectangular area. In this case, the width cannot be greater than half of the available fencing length (225m), as dividing it into two sections requires two equal lengths. Additionally, the width cannot be negative or zero, as it needs to have a positive value. Therefore, the domain of the area function is 0 < w ≤ 225.
To determine the range of the area function, we need to find the possible values for the area. Since the width cannot be negative or zero, the area will also be positive. As the width increases, the area will increase until it reaches a maximum value. Therefore, the range of the area function is all positive numbers.
c) To find the dimensions that give the maximum area, we can use calculus. By taking the derivative of the area function with respect to the width and setting it equal to zero, we can find the critical points.
A'(w) = (450 - 3w)/2 - (w(450 - 3w))/(2(w^2))
= (450 - 3w - 450w + 3w^2)/(2w^2)
= (-450w + 3w^2)/(2w^2)
Setting A'(w) = 0:
(-450w + 3w^2)/(2w^2) = 0
Since the numerator equals zero, -450w + 3w^2 = 0.
Rearranging this equation, we get 3w^2 = 450w.
Dividing both sides by 3w, we get w = 150.
Therefore, the width that gives the maximum area is 150m. To find the corresponding length, we can substitute this value back into the area function equation:
A(w) = (450 - 3w)/(2w)
A(150) = (450 - 3(150))/(2(150))
= 900/300
= 3
Hence, the dimensions that give the maximum area are a width of 150m and a length of 3m.
Part b) To determine the domain of the area function, we need to consider the limitations on the width (w) of the rectangular area. In this case, since we are dividing the area into two sections, the width cannot be greater than half of the total length of the fencing, which is 450m. Therefore, the domain will be 0 ≤ w ≤ 225 (half of 450).
Next, to determine the range of the area function, we need to find the maximum and minimum values that the area can take. Intuitively, we know that the area will be maximized when the width and length of the rectangle are as similar as possible, meaning the rectangle is square-shaped.
Part c) To determine the dimensions that give the maximum area, we can take the derivative of the area function with respect to the width and set it equal to zero. Let's first rewrite the area function to make it clearer:
A(w) = (450 - 3w)w / 2
To find the maximum, we take the derivative of A(w) with respect to w, and then set it equal to zero:
A'(w) = [(450 - 3w)/2] - [3w/2]
= (450 - 3w - 3w) / 2
= (450 - 6w) / 2
= 225 - 3w
Setting A'(w) equal to zero, we get:
225 - 3w = 0
3w = 225
w = 225/3
w = 75
Therefore, a width of 75 meters will give the maximum area. To find the corresponding length (L), we can substitute this value of w into the area function:
A(75) = (450 - 3(75)) * 75 / 2
= (450 - 225) * 75 / 2
= 225 * 75 / 2
= 11250 / 2
= 5625
So, the dimensions that give the maximum area are a width of 75 meters and a length of 225 meters.