We are studying quadratic functions.
Suppose you have 80 feet of fence to enclose a rectangular garden. The function A= 40x-x2 gives you the area of the garden in square feet where x is the width in feet.
What width gives you the maximum gardening area?
To find the width that gives you the maximum gardening area, we need to determine the value of x that maximizes the quadratic function A = 40x - x^2.
To find the maximum value of the quadratic function A, we can use the vertex formula. The vertex formula states that the x-coordinate of the vertex of a quadratic function in the form y = ax^2 + bx + c is given by x = -b/2a.
In our case, the quadratic function is A = -x^2 + 40x. Comparing this with the general form y = ax^2 + bx + c, we have a = -1 and b = 40. Substituting these values into the formula, we have:
x = -b/2a
x = -40 / (2*(-1))
x = -40 / (-2)
x = 20
So, the width that gives you the maximum gardening area is x = 20 feet.
To find the width that gives you the maximum gardening area, you need to find the value of x that corresponds to the maximum point on the graph of the quadratic function A = 40x - x^2.
To determine the width, we need to find the x-coordinate of the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In our case, the quadratic function is A = -x^2 + 40x, which can be written in the form ax^2 + bx + c = 0. Comparing the expression with the general quadratic form, we have:
a = -1
b = 40
Now we can calculate the x-coordinate of the vertex using the formula:
x = -b / (2a)
x = -40 / (2*-1)
x = -40 / -2
x = 20
Therefore, a width of 20 feet will give you the maximum gardening area.