question 1 of 10 derek purchased a 2010 model...

# Explanation: ## Step1: Calculate the monthly interest rate The annual interest rate is $r = 6\%=0.06$, so the monthly interest rate $i=\frac{0.06}{12}=0.005$. The number of months $n = 60$. ## Step2: Use the present - value of an ordinary annuity formula to find the present value of the monthly payments The formula for the present - value of an ordinary annuity is $PV = PMT\times\frac{1-(1 + i)^{-n}}{i}$, where $PMT = 149$. Substitute the values: $PV=149\times\frac{1-(1 + 0.005)^{-60}}{0.005}$. First, calculate $(1 + 0.005)^{-60}=\frac{1}{(1 + 0.005)^{60}}$. Using the formula $a^{-n}=\frac{1}{a^{n}}$, and $(1 + 0.005)^{60}\approx1.3488501525$. So, $(1 + 0.005)^{-60}\approx0.7413721962$. Then, $1-(1 + 0.005)^{-60}=1 - 0.7413721962 = 0.2586278038$. $\frac{1-(1 + 0.005)^{-60}}{0.005}=\frac{0.2586278038}{0.005}=51.72556076$. $PV=149\times51.72556076\approx7607.01$. ## Step3: Calculate the balloon payment The initial cost of the car is $P = 21000$. The present value of the monthly payments is $PV\approx7607.01$. The future - value of the initial cost of the car after 60 months is $F_1=21000\times(1 + 0.005)^{60}$. Since $(1 + 0.005)^{60}\approx1.3488501525$, $F_1=21000\times1.3488501525\approx28325.85$. The future - value of the monthly payments is $F_2=149\times\frac{(1 + 0.005)^{60}-1}{0.005}$. $(1 + 0.005)^{60}-1=1.3488501525 - 1=0.3488501525$. $\frac{(1 + 0.005)^{60}-1}{0.005}=\frac{0.3488501525}{0.005}=69.7700305$. $F_2=149\times69.7700305\approx10395.73$. The balloon payment $B$ is the future - value of the initial cost minus the future - value of the monthly payments. $B = 21000\times(1 + 0.005)^{60}-149\times\frac{(1 + 0.005)^{60}-1}{0.005}$ $B\approx28325.85-10395.73 = 17930.12$. # Answer: C. $17,930.12$