Use the intercept form of the equation of the parabola.
Equation: - x^2+7x Example rectangle: 1* 6 Maximum Area: 3.5* 3.5
Practice makes perfect
We are given a parabola with 3 of its points specified.
Since two of these points are actually x-intercepts, it's reasonable to use the intercept form of the equation of the parabola.
y=a(x-p)(x-q)Here p=0 and q=7 are the x-intercepts, while x=1 and y=6 are the coordinates of the third point (1,6).
6=a( 1- 0)( 1- 7)
Let's find a in order to complete the equation.
The area of the rectangle that is 1 m wide equals 6 m^2, so its length has to be 6 m. To find the dimensions that result in the maximum area we must find the vertex. We can do this using the graphing calculator. We first press the Y= button and type the function in. Having written the function, we can push GRAPH to draw it.
To find the function's maximum value, push 2nd and TRACE and choose the fourth option, maximum. The calculator will prompt us to choose a left-bound, a right-bound, and to provide the calculator with a guess as to where the maximum might be. Be sure to choose the bounds appropriately so that the maximum is between these values.
We can see that the vertex of the parabola lies at (3.5,12.25), which means that area's maximum width is equal to 3.5 meters. To find the second dimension of this rectangle we have to divide the area by its width.
12.25/3.5=3.5
This means that the area of the rectangle is maximal when both dimensions are equal to 3.5.
