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Math  /  Algebra

QuestionWrite an exponential function for the yearly interest rate then rewrite the function to find the equivalent monthly interest rate.
APR stands for annual (yearly) interest rate.
12\% APR
Use the variable tt for time.
Equation for yearly interest rate: A(t)=P(□)A(t)=P(\square)
Equation for monthly interest rate: A(t)=P(□)=A(t)=P(\square)=
Monthly interest rate: □\square

Studdy Solution

STEP 1

1. The annual interest rate is given as 12\% APR.
2. We need to express this as an exponential function for yearly compounding.
3. We need to convert the yearly interest rate to an equivalent monthly interest rate.

STEP 2

1. Write the exponential function for the yearly interest rate.
2. Convert the yearly interest rate to an equivalent monthly interest rate.
3. Write the exponential function for the monthly interest rate.

STEP 3

The annual interest rate of 12\% can be expressed as a decimal for calculations: r=0.12 r = 0.12 .
The exponential function for the yearly interest rate is given by the formula for compound interest:
A(t)=P(1+r)t A(t) = P(1 + r)^t
Substituting the given interest rate:
A(t)=P(1+0.12)t A(t) = P(1 + 0.12)^t

STEP 4

To find the equivalent monthly interest rate, we need to divide the annual rate by 12 (the number of months in a year) and adjust the exponent accordingly.
The monthly interest rate rmonthly r_{\text{monthly}} is:
rmonthly=0.1212=0.01 r_{\text{monthly}} = \frac{0.12}{12} = 0.01
The monthly compounding formula becomes:
A(t)=P(1+rmonthly)12t A(t) = P(1 + r_{\text{monthly}})^{12t}
Substituting the monthly rate:
A(t)=P(1+0.01)12t A(t) = P(1 + 0.01)^{12t}

STEP 5

The equivalent monthly interest rate is 1\% per month.
The exponential function for the monthly interest rate is:
A(t)=P(1+0.01)12t A(t) = P(1 + 0.01)^{12t}
Monthly interest rate:
0.01 0.01
The exponential function for the yearly interest rate is:
A(t)=P(1.12)t A(t) = P(1.12)^t
The exponential function for the monthly interest rate is:
A(t)=P(1.01)12t A(t) = P(1.01)^{12t}
Monthly interest rate:
0.01 0.01

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