The problem involves determining the values of m and c such that the area of a shaded region is 4.5 square units. The context includes the intersection of a linear equation and a quadratic function, and the use of definite integrals to find the area between the curves.
Several participants have provided guidance on calculating intersection points and setting up integrals. There is an ongoing exploration of algebraic manipulation, with some participants noting challenges in solving for m and discussing the nature of the resulting polynomial equation.
Participants mention that the algebra is complex and that there may be extraneous roots in the polynomial equation derived from the area calculation. There is also a reference to the difficulty of achieving integer roots in the context of the problem.
So ## x=-m+3 ##. You integrate from ## x=-1 ## to ## x=-m+3 ##. That should be obvious.gex said:
That is what I'm finding very difficult right now. I've been trying to solve for m myself but can't seem to do it. I used an online calculator and the provided root isn't an integer which leads me to believe I've gone wrong somewhere else in my algebra.Charles Link said:
The algebra is not the easiest. Here's what I got after multiplying both sides by ## -6 ## (originally it was ## -\frac{m^3}{6 }+... ##) : ## \\ ## ## m^3-12m^2+48m-37=0 ##. ## \\ ## See if you can work to that result. ## \\ ## This equation readily factors with one simple root. My guess is that you got something slightly different.gex said:
Charles Link said:
gex said:
Quite honest? I took me two attempts, too.Charles Link said:
