# Modeling a Context from a Verbal Description

### New York State Common Core Math Algebra I, Module 5, Lesson 8

Student Outcomes

• Students model functions described verbally in a given context using graphs, tables, or algebraic representations.

Modeling a Context from a Verbal Description

Classwork

Example 1

Christine has \$500 to deposit in a savings account, and she is trying to decide between two banks. Bank A offers 10% annual interest compounded quarterly. Rather than compounding interest for smaller accounts, Bank B offers to add \$15 quarterly to any account with a balance of less than \$1,000 for every quarter, as long as there are no withdrawals. Christine has decided that she will neither withdraw, nor make a deposit for a number of years.

Develop a model that will help Christine decide which bank to use.

Example 2

Alex designed a new snowboard. He wants to market it and make a profit. The total initial cost for manufacturing setup, advertising, etc. is \$500,000, and the materials to make the snowboards cost \$100 per board.

The demand function for selling a similar snowboard is π·(π) = 50,000 β 100π, where π represents the selling price (in dollars) of each snowboard.
a. Write an expression for each of the following in terms of π.
Demand Function (number of units that will sell)
Revenue [(number of units that will sell)(price per unit, π)]
Total Cost (cost for producing the snowboards)
b. Write an expression to represent the profit.
c. What is the selling price of the snowboard that will give the maximum profit?
d. What is the maximum profit Alex can make?

Exercises

Alvin just turned 16 years old. His grandmother told him that she will give him \$10,000 to buy any car he wants whenever he is ready. Alvin wants to be able to buy his dream car by his 21st birthday, and he wants a 2009 Avatar Z, which he could purchase today for \$25,000. The car depreciates (reduces in value) at a rate is 15% per year. He wants to figure out how long it would take for his \$10,000 to be enough to buy the car, without investing the \$10,000.

1. Write the function that models the depreciated value of the car after π number of years.
a. Will he be able to afford to buy the car when he turns 21? Explain why or why not.
b. Given the same rate of depreciation, after how many years will the value of the car be less than \$5,000?
c. If the same rate of depreciation were to continue indefinitely, after how many years would the value of the car be approximately \$1?
2. Sophia plans to invest \$1,000 in each of three banks.
Bank A offers an annual interest rate of 12%, compounded annually.
Bank B offers an annual interest rate of 12%, compounded quarterly.
Bank C offers an annual interest rate of 12%, compounded monthly.
a. Write the function that describes the growth of investment for each bank in π years.
b. How many years will it take to double her initial investment for each bank? (Round to the nearest whole dollar.)
c. Sophia went to Bank D. The bank offers a βdouble your moneyβ program for an initial investment of \$1,000 in five years, compounded annually. What is the annual interest rate for Bank D?

Lesson Summary

• We can use the full modeling cycle to solve real-world problems in the context of business and commerce (e.g., compound interest, revenue, profit, and cost) and population growth and decay (e.g., population growth, depreciation value, and half-life) to demonstrate linear, exponential, and quadratic functions described verbally through using graphs, tables, or algebraic expressions to make appropriate interpretations and decisions.
• Sometimes a graph or table is the best model for problems that involve complicated function equations

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