Question
Miguel invested $400 in a bank account that pays interest. The account’s value over time, in x years, is given in the table.
A 2-column table with 6 rows. The first column is labeled time (years) with entries 0, 1, 2, 3, 4, 5. The second column is labeled account value (dollar sign) with entries 400, 436, 468, 502, 550, 589.
Which exponential function models the data? Round the numerical values to the nearest hundredth.
f(x) = 1.41(30.69)x
f(x) = 37.74(396.48)x
f(x) = 390.60(1.41)x
f(x) = 401.19(1.08)x
A 2-column table with 6 rows. The first column is labeled time (years) with entries 0, 1, 2, 3, 4, 5. The second column is labeled account value (dollar sign) with entries 400, 436, 468, 502, 550, 589.
Which exponential function models the data? Round the numerical values to the nearest hundredth.
f(x) = 1.41(30.69)x
f(x) = 37.74(396.48)x
f(x) = 390.60(1.41)x
f(x) = 401.19(1.08)x
Answers
GPT 3.5
To determine which exponential function models the data, we can use the formula for compound interest:
A = P(1 + r)^t
where:
A = final account value
P = principal amount (initial investment)
r = interest rate
t = time in years
In this case, we have P = $400, A values from the table, and t values from the table. We can calculate the values of r using the formula:
r = (A / P)^(1/t) - 1
We then take the average value of r from the different combinations of A and t and use that to find the base value in the exponential function.
Let's calculate the values of r:
For t = 1, r = (436 / 400)^(1/1) - 1 = 0.09
For t = 2, r = (468 / 400)^(1/2) - 1 = 0.095
For t = 3, r = (502 / 400)^(1/3) - 1 = 0.1
For t = 4, r = (550 / 400)^(1/4) - 1 = 0.112
For t = 5, r = (589 / 400)^(1/5) - 1 = 0.118
Taking the average, we get r ≈ 0.1048
Now we can find the base value for the exponential function:
f(x) = 400(1 + 0.1048)^x ≈ 400(1.1048)^x
This gives us:
f(x) = 441.92(1.1048)^x
Therefore, the exponential function that models the data is: f(x) = 441.92(1.1048)^x.
A = P(1 + r)^t
where:
A = final account value
P = principal amount (initial investment)
r = interest rate
t = time in years
In this case, we have P = $400, A values from the table, and t values from the table. We can calculate the values of r using the formula:
r = (A / P)^(1/t) - 1
We then take the average value of r from the different combinations of A and t and use that to find the base value in the exponential function.
Let's calculate the values of r:
For t = 1, r = (436 / 400)^(1/1) - 1 = 0.09
For t = 2, r = (468 / 400)^(1/2) - 1 = 0.095
For t = 3, r = (502 / 400)^(1/3) - 1 = 0.1
For t = 4, r = (550 / 400)^(1/4) - 1 = 0.112
For t = 5, r = (589 / 400)^(1/5) - 1 = 0.118
Taking the average, we get r ≈ 0.1048
Now we can find the base value for the exponential function:
f(x) = 400(1 + 0.1048)^x ≈ 400(1.1048)^x
This gives us:
f(x) = 441.92(1.1048)^x
Therefore, the exponential function that models the data is: f(x) = 441.92(1.1048)^x.