Jacqueline has 100 feet of fencing to build a pen at her farm. She writes the function A=w(50−w) to represent the area of a pen with width w.
Determine the vertex of the function and state what it means in the context of the scenario.
To find the vertex of the function, we need to rewrite it in vertex form.
The vertex form of a quadratic function is given by:
A = a(x-h)^2 + k
where (h, k) represents the vertex of the parabola.
In this case, we have:
A = w(50 - w)
Expanding the expression, we get:
A = 50w - w^2
To rewrite this in vertex form, we can complete the square:
A = -(w^2 - 50w)
To complete the square, we need to add (50/2)^2 = 625 to the expression inside the parentheses:
A = -(w^2 - 50w + 625 - 625)
A = -(w^2 - 50w + 625) + 625
A = -[(w - 25)^2 - 625] + 625
A = -(w - 25)^2 + 625
Now that the expression is in vertex form, we can easily identify the vertex.
In this case, the vertex is (25, 625).
In the context of the scenario, the vertex represents the maximum area of the pen that can be enclosed using the given amount of fencing (100 feet). So, the vertex of this function shows that the maximum area can be achieved when the width of the pen is 25 feet, and the area in that case would be 625 square feet.
To find the vertex of the function, we need to rewrite it in the standard vertex form.
The given function is A = w(50 - w)
Expanding the equation, we get A = 50w - w^2
To rewrite it in standard vertex form, we need to complete the square. Let's do that:
A = -w^2 + 50w
To complete the square, we need to add and subtract (50/2)^2 = 625 to the equation:
A = -w^2 + 50w + 625 - 625
= -(w^2 - 50w + 625) + 625
Now, we can rewrite it as:
A = -(w - 25)^2 + 625
The vertex form of the equation is A = a(w - h)^2 + k, where the vertex is at (h, k).
Comparing our equation with the vertex form, we can see that the vertex is at (25, 625).
In the context of the scenario, the vertex (25, 625) represents the maximum area of the pen that can be achieved with the given amount of fencing.