Question

1. bank account x has an initial deposit of $2,000 in week 0 and grows by $200 in each week after that.

bank account y has an initial deposit of $50 in the same week 0 and doubles each week after that.
when will account y have the same amount of money as account x? explain your reasoning.
(from unit 6, lesson 1.)

1. let f be a rule that takes the day of the week as the input and gives a food that you ate that day as the output. for example, f(monday) = chicken.

a. write three sample input - output pairs for f.
b. does each input to f have exactly one output? explain how you know.
c. explain why f is not a function.

Explanation:

Problem 5

Step1: Define formulas for accounts

Account X: linear growth, formula $A_X = 2000 + 200n$ (n = weeks).
Account Y: exponential growth, formula $A_Y = 50 \times 2^n$.

Step2: Set equations equal

Find n where $2000 + 200n = 50 \times 2^n$.

Step3: Test integer values for n

• n=10: $A_X=2000+200(10)=4000$; $A_Y=50×2^{10}=51200$ (Y > X)
• n=9: $A_X=2000+200(9)=3800$; $A_Y=50×2^9=25600$ (Y > X)
• n=8: $A_X=2000+200(8)=3600$; $A_Y=50×2^8=12800$ (Y > X)
• n=7: $A_X=2000+200(7)=3400$; $A_Y=50×2^7=6400$ (Y > X)
• n=6: $A_X=2000+200(6)=3200$; $A_Y=50×2^6=3200$ (Equal!)

We create input - output pairs by assigning a food to each day of the week as per the rule F. For example, we can choose common foods eaten on different days.

A function requires each input to have exactly one output. For rule F, when we consider a specific day (input), we ate only one food (output) on that day (assuming we are talking about a specific instance of eating on that day). So each input (day) has one output (food eaten that day).

Answer:

Account Y equals Account X at week 6.

Problem 6a