a Note that the management fee and babysitting earnings will be paid/deposited after the interest has been earned on the principal.
B
b Subtract the balance of the accounts in Year 0 from the balances in Year 2, then divide by the difference in years.
C
c Subtract the balances of the accounts in Year 3 from the balances in Year 5, then divide by the difference in years.
D
d One account increases exponentially and one increases linearly. Which one will overtake the other?-
A
aTable:
Year
Mateo
Marcy
0
1000
1000
1
1200
1165.00
2
1400
1343.20
3
1600
1535.66
4
1800
1743.51
5
2000
1967.99
B
bMateo: $200 per year
Marcy: $171.60 per year
C
cMateo: $200 per year
Marcy: $216.17 per year
D
d Marcy's account. See solution.
Practice makes perfect
a Let's make one diagram for Mateo and another for Marcy.
Mateo
From the exercise, we know that Mateo starts with $1000 in his savings account. This means that when x=0, we have 1000.
Year
Balance
0
1000
At the end of each year, Mateo's account grows by $200. With this information, we can complete the table for Mateo.
Year
Balance
0
1000
1
1200
2
1400
3
1600
4
1800
5
2000
Marcy
Marcy also started out with $1000.
Year
Balance
0
1000
Her investment earns 8 % interest annually, and she also adds $100 to the account after her interest has been paid. Marcy also needs to pay a management fee of $15 a year. We can use this information to calculate the amount in Marcy's bank account at the end of the first five years.
t
Balance Calculation
≈
1
1000 * 1.08 + 100 - 15
1165
2
1165 * 1.08 + 100 - 15
1343.20
3
1343.20 * 1.08 + 100 - 15
1535.66
4
1535.66 * 1.08 + 100 - 15
1743.51
5
1743.51 * 1.08 + 100 - 15
1967.99
Now we can create the table for Marcy.
Year
Balance
0
1000
1
1165.00
2
1343.20
3
1535.66
4
1743.51
5
1967.99
b To determine the average rate of change of the balances of the accounts from Year 0 to Year 2, we will subtract the balance at Year 0 from the balance at Year 2, then divide by the difference in years.
Year
Mateo
Marcy
0
1000
1000
1
1200
1165.00
2
1400
1343.20
3
1600
1535.66
4
1800
1743.51
5
2000
1967.99
Now we can calculate the average rate of change. Since time is the independent variable and the dollar amount in the account is the dependent variable, we have a unit of dollars per year.
Mateo:& 1400- 1000/2- 0= $200 per year [1.5em]
Marcy:& 1343.20- 1000/2- 0= $171.60 per year
c Let's repeat the procedure from Part B, this time considering Years 3 and 5.
Year
Mateo
Marcy
0
1000
1000
1
1200
1165.00
2
1400
1343.20
3
1600
1535.66
4
1800
1743.51
5
2000
1967.99
Now we can calculate the average rate of change. Since time is the independent variable and the dollar amount in the account is the dependent variable, we have a unit of dollars per year.
Mateo:& 2000- 1600/5- 3= $200 per year [1.5em]
Marcy:& 1967.99- 1535.66/5- 3≈ $216.17per year
d Examining the results from Part B and C, we notice that the average rate of change increases exponentially for Marcy's account, while it remains constant for Mateo. Therefore, Marcy will, in the long run, end up with more money than Mateo. We can see this if we expand the table by just one more year.
