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Chapter 8: Problem 11

In Exercises 11-14, use the formula $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} $$ Round all computations to the nearest dollar. Suppose that you drive 40,000 miles per year and gas averages \(\$ 4\) per gallon. a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SUV averaging 16 miles per gallon? b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays \(5.2 \%\) compounded monthly, how much will you have saved at the end of six years?

Short Answer

Expert verified
By owning a hybrid vehicle, you can save $6000 annually and $500 monthly. Depositing these monthly savings in an annuity paying \(5.2\%\) compound interest could accumulate substantial savings over six years.

Step by step solution

01

Calculation of Annual Fuel Expenses for Hybrid Car and SUV

First, calculate the annual fuel consumption. With the hybrid car consuming 1 gallon per 40 miles, in 40000 miles it would consume \(\frac{40000}{40}=1000\) gallons. Similarly, the SUV consumes 1 gallon per 16 miles and thus in 40000 miles it would consume \(\frac{40000}{16}=2500\) gallons. Multiply the number of consumed gallons with the price per gallon (i.e., $4) to find the total annual cost for both.\The hybrid would cost \(1000*4=4000$), whereas the SUV would cost \(2500 * 4=10000$\).\\Then, to find the annual savings, subtract the cost of the hybrid from the cost of the SUV (\(10000-4000)$, which equates to $6000.
02

Calculate Monthly Fuel Savings

You save $6000 per year by choosing a hybrid over an SUV. To find out the monthly savings, divide the yearly savings by 12. \( \frac{6000}{12} = $500 \) per month.
03

Calculate the Savings from the Annuity

Now, depositing the monthly savings of $500, which were calculated in the previous step, into an annuity for 6 years at a rate r of \(5.2\% = 0.052\) per year compounded monthly (n=12), it can be used:\The formula for the future value of an annuity, \( A = \frac{P[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}} \). \Substitute P = 500, r = 0.052, n = 12, and t = 6.\\( A = \frac{500[(1+\frac{0.052}{12})^{12 * 6}-1]}{\frac{0.052}{12}} \),\After performing the calculations, rounding to the nearest dollar, gives the total savings after 6 years.
04

Summary of Results'

Each year the hybrid car provides a saving of $6000 in fuel expenses compared to the SUV. Monthly this saving is $500. By depositing these savings into an annuity with interest of \(5.2\%\) compounded monthly, the total savings over 6 years could be well above the annual saving. In addition, one would have contributed to lessening the carbon footprint.

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