Your checking account has a constant balance of $500. Let the function m represent the balance of your savings account after t years. The table shows the total balance of the accounts over time.
Year, t Total
balance
0
1
2
3
4
5
$2500
$2540
$2580.80
$2622.42
$2664.86
$2708.16
a. Write a function B that represents the total balance after t years.
The function B represents the total balance after t years. From the given information, we can see that the total balance starts at $500 and increases by a constant rate over time. The pattern suggests exponential growth.
To find the exponential growth function, we can use the equation for compound interest:
B = P(1 + r)^t
Where B is the total balance, P is the initial balance, r is the growth rate as a decimal, and t is the time in years.
In this case, the initial balance P is $500 and the growth rate r is unknown. We can find the growth rate r by dividing the difference in balance after each year by the initial balance:
r = (B - P) / P
Using the given values from the table:
- After 1 year: B = $2500, P = $500
- After 2 years: B = $2540, P = $500
- After 3 years: B = $2580.80, P = $500
- After 4 years: B = $2622.42, P = $500
- After 5 years: B = $2664.86, P = $500
Using the formula for r, we can calculate the growth rate:
r = ($2500 - $500) / $500 = 4
r = ($2540 - $500) / $500 = 4.08
r = ($2580.80 - $500) / $500 = 4.162
r = ($2622.42 - $500) / $500 = 4.24484
r = ($2664.86 - $500) / $500 = 4.32872
It appears that the growth rate is increasing over time. To get an approximation of the growth rate, we can take the average of these values:
r ≈ (4 + 4.08 + 4.162 + 4.24484 + 4.32872) / 5 ≈ 4.162
Plugging in the values into the equation for compound interest:
B = $500(1 + 4.162)^t
Thus, the function B that represents the total balance after t years is:
B(t) = $500(1 + 4.162)^t