judy wants to fence the sides of the yard in front of her house . She bought 60m of fence and wants the max area she can fence in . The quad function a(x)=-2x^2+60x, where x is the width of the yard in metres , represent area enclosed . Determine the max area possible , explain or show how you arrived at your answer .
as with any parabola ax^2+bx+c, the vertex is at x = -b/2a
so, here the maximum is achieved at x = 15.
To determine the maximum area possible for the fenced-in yard, we need to find the value of x that will maximize the quad function a(x)=-2x^2+60x.
To find the maximum value of the quad function, we can use calculus by taking the derivative of the function and setting it equal to zero. Let's start with finding the derivative of a(x):
a'(x) = -4x + 60
Now, let's set the derivative equal to zero and solve for x:
-4x + 60 = 0
-4x = -60
x = 15
So, x = 15 is a critical point. To check if this point is a maximum or minimum, we can take the second derivative of a(x):
a''(x) = -4
Since the second derivative is a constant (-4), we can conclude that x = 15 is indeed a maximum point.
Therefore, to maximize the area of the fenced-in yard, Judy should have a width of 15 meters. Now we can substitute this value of x back into the area formula:
a(x) = -2x^2 + 60x
a(15) = -2(15)^2 + 60(15)
a(15) = -2(225) + 900
a(15) = -450 + 900
a(15) = 450
So, the maximum possible area Judy can fence in is 450 square meters.
To determine the maximum area possible, we need to analyze the quadratic function a(x) = -2x^2 + 60x, where x is the width of the yard in meters.
Step 1: Understand the Problem
Judy wants to fence the sides of the yard in front of her house using 60 meters of fence. Given the quadratic function a(x) representing the enclosed area, we need to find the maximum area possible.
Step 2: Identify the Key Values
In the quadratic function a(x) = -2x^2 + 60x, the coefficient of x^2 is -2, and the coefficient of x is 60.
Step 3: Determine the Vertex
The vertex of a quadratic function in the form a(x) = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = -2 and b = 60.
x = -b / (2a) = -60 / (2(-2)) = -60 / (-4) = 15
Substituting x = 15 into the equation, we can find the corresponding value of a(x):
a(15) = -2(15)^2 + 60(15) = -2(225) + 900 = -450 + 900 = 450
Therefore, the vertex of the function is (15, 450).
Step 4: Determine the Maximum Area
Since the vertex represents the maximum or minimum point of a quadratic function, we have determined that the maximum area possible is 450 square meters.
Step 5: Interpretation
Judy can enclose a maximum area of 450 square meters using the 60-meter fence by making the width of the yard 15 meters. This means she should create a rectangular yard with a width of 15 meters and any length that fits within the fence perimeter to achieve the maximum possible area.