please i need to compare it and correct my solutions.

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1) You work for a cereal company as a box designer. Your job is to find the dimensions of a box. The volume of the box is given by the following:

$v(x) = 8x^{3 } -108x^{2} + 360$

a. Use a graphing calculator to determine the length, width, and height of the box so it will hold the largest possible volume. 
b. Factor the polynomial completely to find the $footnotesize{x}$-intercepts of the graph. In context of this problem, what do the intercepts represent?
c. What is the domain and range for the box?

2) In the summer time your friend wants to have a snow cone stand at a local grocery store parking lot. You want to make sure he actually makes money. The grocery store charges $300 per month. All products are calculated to be 25 cents per customer. Your plans are to sell each cone for $1.25.

a. Write a mathematical expression representing the cost of operation for a month.
b. Write a mathematical expression representing sales.
c. Create an equation that will represent overall profit of the operation for a month.
d. Simplify the equation into slope intercept form and standard form. What is the slope and explain its meaning in the context of selling snow cones? What is the intercept and what is its meaning?
e. Describe the method of graphing the slope intercept form and also the standard form of the equation.
f. What is the domain and range of the equation? Is the relation a function, why?
g. How many snow cones do you need to sell to break even in the first month?
h. How much is the profit if you sell 425, 550 or 700 snow cones in the first month of business?

3) Thomas and Jenny went to a bakery to pick up some cookies for dessert. Thomas bought 3 pumpkin cookies and 2 raisin cookies. Jenny bought 2 pumpkin cookies and 4 raisin cookies. Thomas consumed 870 calories and Jenny consumed 750 calories.

a. Write a system of equations to represent this situation.
b. Solve each equation for $footnotesize{y}$ and graph to find the solution. How many calories are the pumpkin and raisin cookies?
c. Solve the same equations using substitution and the elimination methods. Compare all of your answers.

4) Write a rational expression with at least one $footnotesize{x}$ in the denominator and then analyze it using the following questions:

a. Set your expression equal to 0. Does this expression have any solutions? Why, or why not?
b. Set your expression equal to $footnotesize{y}$ and graph the equation on a graphing utility or on your calculator. Are there any values that $footnotesize{x}$ cannot be? Explain.
c. How do the numbers and variables in your equation relate to the features of the graph?
d. Identify the denominator of your expression; add 6 to the denominator only. How does the graph change?

5) Graph the following three equations on your own device and answer the questions below.
$#1 ,, f(x) = 5x + 7$
$#2 ,, f(x) = x^{2} + 6$
$#3 ,, f(x) = 2^{x} + 3$

a. What is $footnotesize{f(10)}$?
b. What is $footnotesize{f(100) }$?
c. What is $footnotesize{f(1000) }$?
d. Which function has the highest value at each value of $footnotesize{x}$?
e. Use your answers to decide which type of function will increase the fastest?
f. At what point did this function exceed the other two?
g. Do you think this would work for any function of this type? Why?
h. If these 3 equations represent the growth of mucus in a membrane, describe and state the domain and range for each function.

6) The function $footnotesize{f(t)}$ represents water going into a swimming pool with respect to the number of hours $footnotesize{(t)}$ water is flowing in where $footnotesize{t}$ represents time.
$f(t) = t^{2} + 8t + 9$
There is a leak in the pool and it’s losing water at a rate represented by $footnotesize{d(t)}$
$d(t) = t^{2} + 11t + 4$

a. Write a function $footnotesize{w(t)}$ to represent the amount of water in the pool using the two functions.
b. Use the new function to determine if the pool will leak all of the water.
c. If the pool will drain of all water, how much time will it take?
d. Will $footnotesize{f(t)}$ and $footnotesize{d(t)}$ intersect on a graph? Explain what it means if they do.
e. What is the domain of $footnotesize{f(t)}$$footnotesize{d(t)}$, and $footnotesize{w(t)}$. Explain your answer.

7) You are responsible to provide tennis balls for a tournament. You know that a ball bounces like new when it is dropped and it bounces 82% of the previous height. Because of budget restriction for the tournament you need to test some used balls to make sure they bounce like new tennis balls. You drop a ball and it bounces multiple times; each bounce reaches 82% the height of the previous height.

a. Is this sequence geometric or arithmetic? Explain.
b. What are the heights of the first four bounces of a new ball if it is dropped from a height of 10 feet?
c. Write the equation for the sequence?
d. Does this sequence diverge or converge? Explain.
e. What is the sum of the heights of the bounces for the first ten bounces of a new ball if it is dropped from 10 feet?

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