Question

the exponential model below represents the balance of a loan, in dollars, x years from now. what is meaning of 1.05 in the model? f(x) = 8458(1.05)^x show your work here hint: to add an exponent (x^y), type \exponent\ or press \^\. every year, the loans balance increases by 105% the loans balance will be zero after 5 years every year, the loans balance decreases by 105% the loan will have doubled in balance after 5 years every year, the loans balance increases by 5% every year, the loans balance decreases by 5%

Explanation:

Step1: Recall Exponential Growth Formula

The general form of an exponential growth model is \( f(x)=a(1 + r)^x \), where \( a \) is the initial amount, \( r \) is the growth rate (as a decimal), and \( x \) is the time.

Step2: Compare with Given Model

In the given model \( f(x)=8458(1.05)^x \), we compare it to \( f(x)=a(1 + r)^x \). Here, \( 1 + r = 1.05 \).

Step3: Solve for Growth Rate \( r \)

Subtract 1 from both sides: \( r=1.05 - 1=0.05 \). To convert this decimal to a percentage, we multiply by 100: \( 0.05\times100 = 5\% \). Since \( r>0 \), it is a growth rate, meaning the loan's balance increases by 5% each year.

Answer:

Every year, the loan's balance increases by 5%