a person plans to fence a pig pen with 100 meters of fencing. Derive a function that determines the area of the pig pen that can be enclosed by the fencing [A(w)] with respect to the pig pens width w. find the maximum area of the pig pen that can be enclosed by the fencing.
a person plans to fence a pig pen with 100 meters of fencing. Derive a function that determines the area of the pig pen that can be enclosed by the fencing [A(w)] with respect to the pig pens width w. find the maximum area of the pig pen that can be enclosed by the fencing.
plS ANSWER
If the width is w, the length must be 50-w.
So, the area is
a = w(50-w)
The maximum area is achieved when x=25. You can find what it is.
Note that the max area is when the rectangle is a square.
Where did the 50 come from? What exactly does 50 represent?
To derive a function that determines the area of the pig pen enclosed by the fencing, we need to understand the constraints. The perimeter of a rectangular pig pen can be represented by the equation:
P = 2w + 2L
where P represents the perimeter, w represents the width, and L represents the length. In this case, we are given that the perimeter is 100 meters, so we can substitute this value into the equation:
100 = 2w + 2L
We can simplify this equation by dividing both sides by 2:
50 = w + L
Since we are interested in finding the maximum area enclosed by the fencing, we need to express the area in terms of a single variable. The formula for the area of a rectangle is:
A = w * L
To express the area in terms of a single variable, we can use the perimeter equation to solve for L in terms of w:
50 = w + L
L = 50 - w
We can substitute this value back into the area formula:
A = w * (50 - w)
A = 50w - w^2
So the function that determines the area of the pig pen enclosed by the fencing is A(w) = 50w - w^2.
To find the maximum area, we need to find the vertex of the quadratic equation. The vertex represents the maximum point of the graph. The vertex of a quadratic equation in the form of ax^2 + bx + c can be found using the formula:
x vertex = -b / (2a)
For our equation A(w) = 50w - w^2, the vertex can be found using this formula:
w vertex = -(-50) / (2(-1))
w vertex = 50 / 2
w vertex = 25
So the maximum area of the pig pen that can be enclosed by the fencing occurs when the width is 25 meters. We can find the maximum area by substituting this value back into our equation:
A(w) = 50w - w^2
A(25) = 50(25) - (25^2)
A(25) = 1250 - 625
A(25) = 625 square meters
Therefore, the maximum area of the pig pen that can be enclosed by the fencing is 625 square meters.