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Chapter 0: Problem 108

Semiannual Compound Assume that a \(\$ 1000\) investment earns interest compounded semiannually. Express the value of the investment after 2 years as a polynomial in the annual rate of interest \(r\).

Short Answer

Expert verified
The polynomial is \[1000 + 2000r + 1500r^2 + 500r^3 + 62.5r^4\]

Step by step solution

01

Understand Compounding Semiannually

Semiannual compounding means interest is applied twice a year. For an annual rate of interest, the rate for each period (6 months) will be half of the annual rate.
02

Determine Periods and Rate per Period

If the annual interest rate is denoted as \(r\), the semiannual rate per period is \(\frac{r}{2}\). Over 2 years, there are 4 semiannual periods.
03

Calculate the Value After Each Period

The value of the investment compounded over each period can be found using the formula: \[ A = P \left(1 + \frac{r}{2}\right)^n \]where \(P\) is the principal amount, \(r\) is the annual interest rate, and \(n\) is the number of periods.
04

Substitute Values into the Formula

Here, \(P = 1000\), \(r = r\), and \(n = 4\). Substituting these values, we get:\[ A = 1000 \left(1 + \frac{r}{2}\right)^4 \]
05

Expand the Polynomial

Expand \(\left(1 + \frac{r}{2}\right)^4\) using the binomial theorem:\[\left(1 + \frac{r}{2}\right)^4 = 1 + 4 \cdot \frac{r}{2} + 6 \cdot \left(\frac{r}{2}\right)^2 + 4 \cdot \left(\frac{r}{2}\right)^3 + \left(\frac{r}{2}\right)^4= 1 + 2r + \frac{6r^2}{4} + \frac{2r^3}{4} + \frac{r^4}{16}= 1 + 2r + \frac{3r^2}{2} + \frac{r^3}{2} + \frac{r^4}{16}\]
06

Polynomial Form

Multiply the expanded polynomial by 1000 to express as a polynomial in \(r\):\[A = 1000 \left(1 + 2r + \frac{3r^2}{2} + \frac{r^3}{2} + \frac{r^4}{16}\right)= 1000 + 2000r + 1500r^2 + 500r^3 + 62.5r^4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compounded Interest
Understanding compounded interest is crucial for financial literacy. Compounded interest means that interest is added not only to your initial investment but also to the interest that accumulates over time. This results in exponential growth of your investment rather than linear growth. In this exercise, interest is compounded semiannually, which means it is added twice a year.

For instance, if you invest \( \$1000 \) at an annual interest rate of 4\% compounded semiannually, the interest is calculated and added every six months. So, instead of getting 4\% annually, you will get \( 2\% \) every six months. This accelerates the growth of your investment.
Annual Interest Rate
The annual interest rate, often represented as \( r \, \), is the percentage of the principal earned as interest in one year. In this exercise, the annual interest rate is divided by two because the interest is compounded semiannually.

Mathematically, for an annual interest rate of \( r \), the semiannual interest rate becomes \(\frac{r}{2}\). Over two years, there are four semiannual periods. This is a critical step in calculating the compounded interest.

More generally, if the annual interest rate is 6\%, then the semiannual rate would be 3\%, and it is applied every six months. This frequent compounding period leads to a higher final amount compared to simple annual interest.
Binomial Theorem
The binomial theorem is a powerful formula for expanding expressions that are raised to a power. It is used in this exercise to expand \( \frac{r}{2} \).

According to the binomial theorem, \( (a+b)^n \) can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k \). In our case, \( (1 + \frac{r}{2}) ^ 4 \) needs to be expanded.

By applying the binomial theorem, we get: \( 1 + 4 \frac{r}{2} + 6 \frac{r^2}{4} + 4 \frac{r^3}{8} + \frac{r^4}{16} \). This simplifies to \( 1 + 2r +1.5r^2 + 0.5r^3 + \frac{r^4}{16} \). The resulting expanded form helps in transforming our initial problem into a polynomial.
Polynomial Expansion
Polynomial expansion is the process of expressing a polynomial raised to a power as the sum of its individual terms. The expanded polynomial represents the value of the investment, accounting for each term's contribution to compounded interest.

In the final step, the expanded polynomial \( (1 + 2r + \frac{3r^2}{2} + \frac{r^3}{2} + \frac{r^4}{16}) \) is multiplied by 1000 to get the actual value of the investment after 2 years:
\[ A = 1000 (1 + 2r + 1.5r^2 + 0.5r^3 + \frac{r^4}{16}) \]
Expanding this gives:
\[ A = 1000 + 2000r + 1500r^2 + 500r^3 + 62.5r^4 \]
This polynomial shows how your investment grows with increasing powers of r, reflecting different sources of target income.

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Most popular questions from this chapter

During the first \(\frac{1}{2}\) hour, the employees of a machine shop prepare the work area for the day's work. After that, they turn out 10 precision machine parts per hour, so the output after \(t\) hours is \(f(t)\) machine parts, where \(f(t)=10\left(t-\frac{1}{2}\right)=10 t-5, \frac{1}{2} \leq t \leq 8\). The total cost of producing \(x\) machine parts is \(C(x)\) dollars, where \(C(x)=.1 x^{2}+25 x+200\). (a) Express the total cost as a (composite) function of \(t\). (b) What is the cost of the first 4 hours of operation?

A cellular telephone company estimates that, if it has \(x\) thousand subscribers, its monthly profit is \(P(x)\) thousand dollars, where \(P(x)=12 x-200\). (a) How many subscribers are needed for a monthly profit of 160 thousand dollars? (b) How many new subscribers would be needed to raise the monthly profit from 160 to 166 thousand dollars?

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\left(-32 y^{-5}\right)^{3 / 5}\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\frac{1}{x^{-3}}\)

Revenue from Sales A store estimates that the total revenue (in dollars) from the sale of \(x\) bicycles per year is given by the function \(R(x)=250 x-.2 x^{2}\). (a) Graph \(R(x)\) in the window \([200,500]\) by \([42000,75000] .\) (b) What sales level produces a revenue of \(\$ 63,000 ?\) (c) What revenue is received from the sale of 400 bicycles? (d) Consider the situation of part (c). If the sales level were to decrease by 50 bicycles, by how much would revenue fall? (e) The store believes that, if it spends \(\$ 5000\) in advertising, it can raise the total sales from 400 to 450 bicycles next year. Should it spend the \(\$ 5000\) ? Explain your conclusion.
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