Discussion Based Assessment 01.07

Part 1
1. The volumeís function is 4(x-30)(x-10)x=8000/27(7√7-10, so the maximum volume of this box will be approximately 2524.52 cm^2.
2. Amplitude is 3 and the period is pi. It has x-axis symmetry. The first positive x intercept is at x=0.856.
3. F(x)=x^4 and g(x)=x^(1/4)
F(g(x))=( x^(1/4))^4=x
G(f(x))=(x^4)^(1/4)=│x│
Y = x is not the same function as y=│x│.
So our f(g(x)) does not equal to g(f(x)).
4. Horizontal Asymptote- [(x^2+3x+2)/(x^2+5x+4)] -> 1 as x-> +_ ∞
Vertical asymptote- [(x^2+3x+2)/(x^2+5x+4)] ->+_∞ as x-> -4
Roots = -2
This function is not definite at x=-4 and -1. At x=-4 this is a vertical asymptote. The point (-1;1/3)there will be a hole in the graph.
There is no zero at -1, because you cannot divide by zero.
5. There can be more x-intercepts than roots because the graph of the function can cross the x axis in the same point twice but it will only count as one root. An example of this could be y=(x-2)^3*(9+3x)/(2x-4)(x+3)

Part 2
I have had trouble with some of the concepts, but what tripped me up the most was the classic box problem. I really had a hard time understanding what the problem was. The wording really tripped me up. I also donít have an actual graphing calculator, and I couldnít open Study Forge to see how to solve the problem and Google just gave me really odd search results.
What confused me most about the problem, especially the wording, was that I thought that we were supposed to imagine cutting the piece of cardboard into individual squares and build a box out of them. After initially solving the problem incorrectly, I contacted my teacher and explained my problem with Study Forge. Once I had access I reviewed how the problem was supposed to be solved and I realized that you were supposed to cut out the corners of the board. Once I realized that it was smooth sailing from there on out, mostly.
The problem was I had to find out the maximum volume of a box that would be made from a piece of cardboard with the corners cut out. The real word applications of this problem are quite applicable in almost daily life. One might need to know the maximum volume of a homemade box for a DIY project. Or maybe they could need to explain how to make a box out of one piece of cardboard.