Assignment 12

by Jeff Hall

Problem 1

Construct a graph of any function y=f(x) by generating a table of values with

the x values in one column and the y values in another.

For my equation, I chose f(x)=x^2, with values of x from -10 to 10.

And the graph:

Problem 2

Here is the spreadsheet of my parametric equation:

And here is my graph:

Problem 4

Generate a Fibonacci sequence using a spreadsheet.

Problem 4a

Here is my spreadsheet of the Fibonacci sequence, from n=1 to 25.

Column C shows the adjacent ratio between consecutive Fibonacci numbers.

Column D shows the ratio between every second Fibonacci numbers.

Notice that the ratio becomes constant as n increases for both ratios, albeit at a difference of 1.

Problem 4b

Construct a Lucas Sequence, where f(1)=1 and f(2)=3

Here is the spreadsheet of the sequence:

Notice that both ratios reach the same limit as the original Fibonacci sequence.

Problem 5

Explore problems of growth, such as those involving savings with interest compounded.

The concept of Time Value of Money (TVM) is used as a reason for people to invest and save

money at the earliest possible age. The concept is relatively simple: Money in an interest-bearing account

will grow exponentially in comparison to the principal invested over time. The more time you keep

money invested, the greater the potential for growth.

Let's use a spreadsheet to illustrate this idea. Suppose you invested \$1,000 in a savings account

earning 3% a year on your 30th birthday. If you never add or take money from the account, how

much money would you have when you turn 65 years old?

The initial investment of \$1,000 has grown to \$2,899 dollars.

This is an increase of 290% from an account growing only 3% a year!

Imagine if you invested in a mutual fund earning an average of 10% a year. You also

increased your investment amount to \$1,000 a year for 10 years, until your 40th birthday.

How much money would you have at age 65?

Wow! Your \$10,000 investment grew 2089%, earning you almost \$200,000.

To see how powerful compounding interest is, imagine that you didn't begin investing until age 40.

To catch up, you decide to invest \$2,000 a year until you reach the age of 65.

Here's the graph:

Amazing. Even after investing \$50,000, or 5 times as much, you earned less than \$190,000.

That is still a great return, but it illustrates nicely the power of compounding interest or TVM.
Problem 6

Explore problems of maximization such as the lidless box formed from a 5x8 sheet

with a square removed from each corner.

Spreadsheets make it easy to try a lot of different options.

In my spreadsheet below, I constructed equations based on the variable of the cut square.

In column A, I input a full range of possible cut squares, from sides equalling zero

to sides equalling 2.5, which would wipe out the entire length of the sheet since

two squares are cut from each side of the original 5x8 sheet.

I then subtracted twice the amount of the square from both the length and width.

Finally, I calculated the volume of the box as Length x Width x Height (Side of Cut Square).

As you can see, the maximum value occurs when the Height equals 1.

Problem 7

Place four numbers in the first row as follows:

A B C D

For each successive row replace the entries by the absolute value of the difference of the entry

just above and the entry just to the right in the previous row. In the fourth position

use the absolute value of the difference of the fourth and first (i.e. cycle)

|A-B| |B-C| |C-D| |D-A|

Will the process lead to 0 in all entries for some row?

Here is my spreadsheet with the numbers 2, 5, 17, and 33:

The process lead to a 0 in all entries by the 9th row.

What is the largest number of rows before a zero row is generated?

The most I was able to get was 11 rows

Problem 8

Use the following exploration to generate a function to predict observed data.

a.Take a cup of hot water and measure its initial temperature (time = 0) and then record temperature readings each minute for 30 minutes. Make note of the room temperature . . .

b. Enter the data on a spread sheet and construct a function that will model the data.

c. Using the function predict the temperature after 45 minutes, 60 minutes, or 300 minutes.

d. Calculate a measure of the error between your model and the observed data by taking the square of the difference for each time, sum the squares, and divide by the number of data points. You can use this statistic to guide refinement of your function to model the data.

The boxed-in portion was given in the problem. The function column

was created by me using the equation y=140e^(-.033*A2)+72.

Notice that this function incorporates the correct time for each row.

Rows 33-35 show the estimated temperature after 45, 60, and 300 minutes, respectively.

The function I created in column C shows the estimates. Notice that the temperature

reaches room temperature after 300 minutes, something we would expect from a good function.

Finally, cell D37 shows the Error of my function. 12.49 is decent.

Here is the graph of the actual temperature and the temperature generated by my function:

Notice that my purple function line is close to the blue actual line.