1. calculus

These problems involve advanced Algebra II problems. I am 100% clueless on all four of these problems... I am in Algebra II honors, top of the class.

I would VERY much appreciate it, if you uploaded your work via a scanner or w/e, so I can view it. If you can't do that, I need you to explain your answers.

Problems:

7. If 1200 cm^2 of material is available to make a box with a square base and an open top. Find the largest possible volume of the box. Use the following steps:
a.) Draw a picture
b) Find the volume in terms of x and h
c) Find the surface area in terms of x and h. Then solve for h in terms of x.
d) Find the volume in terms of x
e) To find max, solve the following equation derived from calculus: 1200-3x^2=0
f) Find h
g) Find V

Here is the other three problems in image form:

2. 7. b) $V = x^2h$ Simply base area times height.
c) $S = x^2+4xh$ Base area plus 4 times the side area.
d) $h = \frac{S-x^2}{4x}$ Isolate h taking care of priority of operations.
e) $V = x^2h=x^2\frac{S-x^2}{4x} = x(S-x^2)$ Replace in V.
$\frac{dV}{dx} = 0 = S-3x^2$ Solve for x.
$x = \sqrt{\frac{S}{3}}$
You should be able to do the rest.

3. 10 : Put divide by $4\pi$ to get $\sqrt{r^2-x^2} -\frac{x^2}{\sqrt{r^2-x^2}}+x=0$ Then multiply by $\sqrt{r^2-x^2}$ to get $r^2-x^2 -x^2 +x\sqrt{r^2-x^2}=0$
$r^2-2x^2= -x\sqrt{r^2-x^2}$ Square both sides $r^4-4x^2r^2 + 4x^4= x^2r^2-x^4$
$r^4-5x^2r^2 + 5x^4= 0$
Put $x^2 = u$ and use the formula to solve quadratic equations.
Exercice #11 is pretty much like #7. You should be able to work it out once you've understood #7.
12 : g o f = g(f(x)) and f o g = f(g(x)) Use either f(x) or g(x) as variable.
e.g. $g(f(x)) = \frac{x+\frac{1}{x}+1}{x + \frac{1}{x}+2}$ Simplify by multiplying by $\frac{x}{x}$.

4. See if this helps you with #7. It is a Word document. Good luck! Molly

square open top box.doc