1) An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.

a) Find the function V that represents the volume of the box in terms of x.
Answer:
You are cutting an x out of each side so the dimensions of the bottom of the box are (6-2x) and (4-2x). Imagine folding up the corners, that will be the height of the box, x
The volume of the box is l*w*h = (6-2x)(4-2x)x

b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here.

Keeping in mind x can't be greater than 2, otherwise the width of the box would be less than 0, the graph is shown below:

See attached

c) Using the graph, what is the value of x that will produce the maximum volume?
Answer. Using the graph, to find the max.. only between 0 and 2, we get x = .78 ft

2) The volume of a cylinder (think about the volume of a can) is given by V = πr^2h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 121 cubic centimeters.

a) Write h as a function of r. Keep "π" in the function's equation.
Answer: h = V/ πr^2

b) What is the measurement of the height if the radius of the cylinder is 3 centimetres? Round your answer to the hundredth's place.
Answer: h = V/ πr^2 = 121/ π*3^2 = 4.28 cm

c)Graph this function. See attached

3) The formula for calculating the amount of money returned for deposit money into a bank account or CD (Certificate of Deposit) is given by the following:

A is the amount of returned
P is the principal amount deposited
r is the annual interest rate (expressed as a decimal)
n is the compound period
t is the number of years

Suppose you deposit $20,000 for 3 years at a rate of 8%.

a) Calculate the return (A) if the bank compounds annually (n = 1).
Answer: A = P(1+r)n = 20,000(1+0.08)3 = $25194.24

b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer: A = P(1+r/4)n*4 = 20,000(1+0.02)^12 = $25364.84

c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
Answer: A = P(1+r/12)^n*12 = 20,000(1+0.08/12^)36 = $25404.74

d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.
Answer: A = P(1+r/365)^n*365 = 20,000(1+0.08/365)^365*3 = $25424.31

e) What observation can you make about the size of increase in your return as your compounding increases more frequently?
Answer: The size of increase in return becomes higher and higher as my compounding increases more frequently.

f) If a bank compounds continuous, then the formula becomes simpler, that is
where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding. Round your answer to the hundredth's place.
Answer: A = Pert = 20,000*e^0.08*3 = $25424.98

g) Now suppose, instead of knowing t, we know that the bank returned to us $25,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth's place.
Answer: A = Pe^rt
Or, A/P = e^rt
Therefore, rt = ln(A/P), or t = ln(A/P)/r = ln(25000/20000)/0.08 = 2.80 years

h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place.
Answer: t = ln(A/P)/r = ln(2)/0.08 = 8.66 years

4) For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, let r = 8%, P = 1, and n = 1 and give the coordinates (t,A) for the points where t = 0, 1, 2, 3, 4. Round the A value to the tenth's place.

a) Show coordinates in this space.

For r = 0.08, P = 1, n = 1, A = 1(1+0.08/1)^(1*t) = 1.08^t
The calculations of the ordered pair (t, A) are shown below.

b) Show graph here.

See attached

5) Logarithms:
a) Using a calculator, find log 1000 where log means log to the base of 10.
Answer: log(1000) = 3

b) Most calculators have 2 different logs on them: log, which is base 10, and ln, which is base e. In computer science, digital computers are based on the binary numbering system which means that there are only 2 numbers available to the computer, 0 and 1. When a computer scientist needs a logarithm, he needs a log to base 2 which is not on any calculator. To find the log of a number to any base, we can use a conversion formula as shown here: log b a = log a/log b

Using this formula, find log _2 1000

Using this formula, find . Round your answer to the hundredth's place.
Answer: log 2 1000 = log 1000 / log 2 = 3/ 0.30103 = 9.965784