8B Retirement & Business Decisions QRP Worksheet
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School
Brigham Young University, Idaho *
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Course
108
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
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MATH 108X Lesson 8B Class Activity: Functions and the Quantitative Reasoning Process
Name
: _______________________________________________
Case Study 1
Retirement Goals
Lee is 26 and recently graduated from college. He is setting goals for retirement. He wants to retire at the age of
65 with a retirement fund from which he can draw an income of $150,000 a year, for the rest of his life. He wants to know how much he needs to save each month to reach his goal. He is also worried about how inflation will affect his retirement savings.
What does it mean to have a retirement fund from which he can draw an income of $150,000 each year of retirement? ANS:
It means that Lee wants to have a savings account or investment that generates $150,000 in annual income during his retirement years, allowing him to cover his living expenses.
Think of at least 3 variables
that Lee should consider.
1.
_ Rate of return on all of his investments.
2.
_. His expected life expectancy.
3.
_. How much he already rate of return oon his retirement savings.
To help simplify their decision Lee make the following assumption: •
Inflation will stay close to 3%
•
He found a 401K retirement account with an APR of 6%, compounded month. He assumes he will keep this APR.
Think of at least 2 more assumptions Lee could make.
1.
_ What age does he want to retire
2.
_ annual income growth.
After retirement he wants to live off of the interest from the account. Figure out an equation to find the Principal, P, if Lee wants the yearly interest of his retirement account to be $150,000.
Equation:
Then use the PMT function in Excel
to find how much he need to save each month to reach his goal.
Monthly PMT=____$2,500,000__________
Now he wants to know how inflation will affect his retirement savings.
He finds the exponential function:
buying power
=
Amount
/(
1
+
rateof inflation
)
years
Use the equation to find out how an average inflation rate of 3% will affect Lee’s yearly retirement amount of $150,000.
Buying power
=
¿
Lee’s $150,000 a year will have the same buying power in the future that ______$47,363___________
(the above amount) does right now. What do you think Lee should he do? How much should he save?
Lee should start saving a monthly amount as calculated using the Excel PMT function to reach his goal. He should regularly review his investment performance and adjust his contributions as needed.
Lee should evaluate and re-evaluate his decisions about retirement many times through the next 39
years. Why is it important and smart that he is starting NOW to think about retirement and start saving? Starting early allows Lee to take advantage of compounding, where his money can grow over time. It also
reduces the pressure of saving a large sum later in life and ensures he has more financial security in retirement.
Case Study 2
Maximizing Profits
Marti and Val are artists. They create their art in the same studio and purchase supplies together. They spent last summer selling their art works at art festivals. They also kept track of their sales. They want to figure out how many works to create and the best way to price their art this summer to maximize their profits.
What two things do Marti and Val want to know? 1.
They want to know how many art works to create 2.
the best way to price their art to maximize profits.
Think of at least 3 variables
that Marti and Val should consider.
1.
_ price per artwork
2.
_number of artworks created and sold
3.
_production costs per artwork
To help simplify their decision they make the following assumptions: •
This summer’s sales will be similar to last summer.
•
Last summer their production cost, including travel, was about $15.50 per art work. This summer the costs should be similar.
Think of at least 2 more assumptions they could make.
1.
_ the demand for their art will remain consistent
2.
_their production costs for the summer will be the same as last summer
Marti and Val
changed their prices several times and collected data.
Use Excel to create a scatterplot to see if there is a relationship between the price and the number of works sold.
Place a linear trendline on your scatterplot and find the equation of the line:
Put the equation here:_________ y= -2.0708X+103.12_______
Show
Average Price Works sold
Artisphere
$25
60
Old Town Arts Fair
$30
41
Art at the Lake
$17
68
Riverwalk Art
Festival
$20
65
Art at the Park
$16
69
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State Street Art Fair
$20
57
Art & Apples
$18
63
Summer Art Fair
$40
16
Downtown Arts Night
$50
1
Western Arts & Crafts
$35
30
Marti and Val
use their linear function to set up a profit function. Profit
=
Price
∗
workssold
−
productioncost
∗
works sold.
They estimate their production cost, including travel, to be about $15.50 per art work. Remember the variable x
is the price of an art work, and works sold is the linear equation you found above.
Add numbers and variables to their profit function:20
What kind of function is the profit function?-LINEAR FUCTION
Find the x-coordinate of the vertex to find the optimum selling price for an art work:
Find the y-coordinate of the vertex to find Marti and Val’s expected profit:
Use your linear function to find how many art works do you think Marti and Val should create:
What should they charge for an art work?
What should they expect in profits?
After the first five art fairs, Marti and Val’s profits were close to their predicted profits. However, at the State Street Art Fair the weather was stormy and they sold very little. They now have excess inventory and only 4 art fairs left.
What do you think they should do?
Discount the prices of their art to sell the excess inventory quickly.
Explore alternative sales channels or online platforms to reach a wider audience.
Adjust their production for the remaining art fairs to avoid overproduction.
Analyze the impact of future weather conditions and local events on sales to better plan for similar situations.
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