Question

activity 3: which is growing faster?
noah is depositing money in his account every week to save money. the graph shows the amount he has saved as a function of time since he opened his account.
elena opened an account the same day as noah. the amount of money e in her account is given by the function e = 8w + 60, where w is the number of weeks since the account was opened.

1. who started out with more money in their account? explain how you know.
2. who is saving money at a faster rate? explain how you know.
3. how much will noah save over the course of a year if he does not make any withdrawals? how long will it take elena to save that much?

Explanation:

Step1: Find Noah's initial amount

From the graph, at $w=0$, Noah's amount is $\$60$.

Step2: Find Elena's initial amount

For Elena's function $E=8w+60$, substitute $w=0$:
$E=8(0)+60=60$

Step3: Compare initial amounts

Both start with $\$60$, so they have the same initial amount.

Step4: Find Noah's weekly rate

From the graph, between $w=0$ and $w=2$, amount goes from 60 to 70. Rate = $\frac{70-60}{2-0}=5$ dollars/week.

Step5: Find Elena's weekly rate

Elena's function is in $y=mx+b$ form, where $m=8$ dollars/week.

Step6: Compare savings rates

$8>5$, so Elena saves faster.

Step7: Calculate Noah's 1-year savings

1 year = 52 weeks. Noah's function: $N=5w+60$. Total amount at 52 weeks:
$N=5(52)+60=260+60=320$
Total saved = $320-60=260$ dollars.

Step8: Find Elena's time to save $\$260$

Elena's total savings = $E-60=8w$. Set $8w=260$:
$w=\frac{260}{8}=32.5$ weeks

Answer:

1. Both Noah and Elena started out with the same amount of money ($\$60$). Noah's initial amount is seen at the $w=0$ point on his graph, and Elena's initial amount is the constant term in her linear function.
2. Elena is saving money at a faster rate. Noah saves $\$5$ per week (calculated from his graph's slope), while Elena saves $\$8$ per week (the coefficient of $w$ in her function).
3. Noah will save $\$260$ over a year. It will take Elena 32.5 weeks to save that much.