Read The Following Instructions To Complete This Task ✓ Solved

Read The Following Instructions In Order To Complete This Discussion

Read The Following Instructions In Order To Complete This Discussion

Read the following instructions in order to complete this discussion, and review the example of how to complete the math required for this assignment: Think of something you want or need for which you currently do not have the funds. It could be a vehicle, boat, horse, jewelry, property, vacation, college fund, retirement money, or something else. Pick something which cost somewhere between $2000 and $50,000. On page 270 of Elementary and Intermediate Algebra you will find the “Present Value Formula,” which computes how much money you need to start with now to achieve a desired monetary goal. Assume you will find an investment which promises somewhere between 5% and 10% interest on your money and you want to purchase your desired item in 12 years.

(Remember that the higher the return, usually the riskier the investment, so think carefully before deciding on the interest rate.)

State the following in your discussion: The desired item, how much it will cost in 12 years, the interest rate you have chosen to go with from part b. Set up the formula and work the computational steps one by one, explaining how each step is worked, especially what the negative exponent means. Explain what the answer means. Does this formula look familiar to any other formulas you are aware of? If so, which formula(s) and how is it similar? Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing: Power, Reciprocal, Negative exponent, Position, Rules of exponents. Your initial post should be words in length.

Sample Paper For Above instruction

To plan for a future purchase of a valuable item, such as a new car, I need to determine how much money I must invest today to have enough funds in 12 years, considering a specific interest rate. I choose to buy a car that currently costs $15,000, with plans to purchase it in 12 years. To achieve this, I will assume an annual interest rate of 7%, which falls within the suggested range of 5% to 10%. Using the Present Value Formula, I can calculate the initial investment needed now.

The Present Value Formula is expressed as:

\[ PV = \frac{FV}{(1 + r)^n} \]

Where PV is the present value or initial investment, FV is the future value or cost of the item, r is the annual interest rate in decimal form, and n is the number of years. Substituting the known values gives:

\[ PV = \frac{15,000}{(1 + 0.07)^{12}} \]

First, I calculate the expression inside the parentheses:

\[ 1 + 0.07 = 1.07 \]

This is the base of the power. Next, I work through the exponent of 12, representing 12 years:

\[ 1.07^{12} \]

Using the rules of exponents, this expression signifies multiplying 1.07 by itself 12 times. To compute this efficiently, I may use a calculator or logarithmic rules. The outcome of this power calculation is approximately 2.2523.

Next, I understand that raising a number to a higher power indicates multiplying it repeatedly, embodying the concept of exponential growth. The reciprocal of this number, which is 1 divided by 2.2523, equals approximately 0.4442. This reciprocal indicates how much the original amount is discounted back to present value.

Therefore, I compute the present value (PV) as:

\[ PV = 15,000 \times \frac{1}{2.2523} \approx 15,000 \times 0.4442 = 6,663 \]

This means I need to initially invest approximately $6,663 today at 7% interest to amass $15,000 in 12 years to buy my car. The negative exponent in the formula indicates the reciprocal or inverse of the growth factor, demonstrating the inverse relationship between future value and present value. It embodies the essence of exponential decay when calculating present worth.

This formula resembles the compound interest formula used to calculate future value, but rearranged to find the present value. Both employ powers and rules of exponents, reflecting the exponential nature of growth or decay in investments.

References

  • Larson, M., & Hostetler, R. (2016). Elementary and Intermediate Algebra. Cengage Learning.
  • Franklin, M., & Smith, J. (2018). Financial Mathematics: Applications and Theory. Wiley.
  • Ross, S. (2011). An Introduction to Mathematical Economics. Springer.
  • Swokowski, E. W., & Cole, J. A. (2011). Algebra and Trigonometry. Brooks Cole.
  • Hogg, R. V., & Tanis, E. (2008). Probability and Statistical Inference. Pearson Education.
  • Investopedia. (2020). Present Value (PV). https://www.investopedia.com/terms/p/presentvalue.asp
  • Statistics How To. (2022). Power Rule in Exponents. https://www.statisticshowto.com/probability-and-statistics/exponents-power-rule/
  • Math is Fun. (2023). Exponents. https://www.mathsisfun.com/exponent.html
  • Department of Finance. (2019). Time Value of Money. https://www.finance.gov/topics/time-value-money
  • Udemy. (2021). Basics of Compound Interest. https://www.udemy.com/course/compound-interest

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