A bank account earning annual compound interest was opened, and no additional deposits or withdrawals were made after the initial deposit. The balance in the account after x years can be modeled by b(x)=625(1.015)x.
The balance in the account increases at what rate per year? Answer by percent, but do not use the percent symbol, so if the rate is 6.7% type 6.7.
The balance in the account after x years is given by the function b(x)=625(1.015)^x.
To find the rate at which the balance increases per year, we can take the derivative of b(x) with respect to x:
b'(x) = 625 * (ln(1.015)) * (1.015)^x
The derivative represents the rate of change of the balance with respect to time. In this case, it represents the rate at which the balance increases per year.
To express this rate as a percentage, we can multiply b'(x) by 100:
rate = 100 * 625 * (ln(1.015)) * (1.015)^x
However, we can simplify this expression further. The value of ln(1.015) is approximately 0.0149. Plugging this in, we get:
rate ≈ 100 * 625 * 0.0149 * (1.015)^x
Simplifying this expression, we find:
rate ≈ 93.125 * (1.015)^x
Therefore, the balance in the account increases at a rate of approximately 93.125 * (1.015)^x percent per year.
what is the %
The rate at which the balance in the account increases per year can be expressed as approximately 93.125 * (1.015)^x percent.
that is incorrect
Apologies for the error. Let's go through the calculation again:
The rate of increase in the balance per year can be found by taking the derivative of the expression b(x) = 625(1.015)^x with respect to x.
b'(x) = 625(ln(1.015))(1.015)^x
The derivative represents the rate of change of the balance with respect to time (in this case, in years). To express this rate as a percentage, we can multiply b'(x) by 100:
rate = 100 * 625 * (ln(1.015))(1.015)^x
The natural logarithm ln(1.015) is approximately 0.0149.
rate ≈ 100 * 625 * 0.0149 * (1.015)^x
Simplifying this expression further, we find:
rate ≈ 93.125 * (1.015)^x
Therefore, the balance in the account increases at a rate of approximately 93.125 * (1.015)^x percent per year.