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Chapter 0: Problem 57

Suppose that \(\$ 5000\) is invested at \(8 \%\) interest, compounded semi annully, for \(t\) years. a) The amount \(A\) in the account is a function of time. Find an equation for this function. b) Determine the domain of the function in part (a).

Short Answer

Expert verified
The amount function is \(A = 5000 (1.04)^{2t}\). Domain is \(t \geq 0\).

Step by step solution

01

identify formula

Use the compound interest formula: equestion: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where: - \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (the initial amount of money), which is \(\$5000\).- \(r\) is the annual interest rate (decimal), which is \(0.08\) (or \(8 \%\)).- \(n\) is the number of times that interest is compounded per year, which is 2 (semiannually).- \(t\) is the number of years.
02

Substitute values

Plug the given values into the formula: \[ A = 5000 \left(1 + \frac{0.08}{2}\right)^{2t} \]
03

Simplify the equation

Simplify inside the parenthesis: \[ A = 5000 \left(1 + 0.04\right)^{2t} \]Combine terms: \[ A = 5000 \left(1.04\right)^{2t} \] This will be the function for the amount in the account as a function of time.
04

Determine the domain

Since the time \(t\) represents the number of years and the number of years cannot be negative, the domain is all non-negative real numbers: \[ DOM(A) = \{ t \in \mathbb{R} \mid t \geq 0 \} \]

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Key Concepts

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

Principal Amount
The principal amount is the initial sum of money that you invest or save. In this exercise, the principal amount is \$5,000. It represents the starting point of your investment before any interest is added. The principal amount is denoted as \(\text{P}\) in the compound interest formula. Understanding your principal amount is crucial, as it significantly impacts the final amount accumulated over time.

  • The larger the principal, the more potential for earning interest.
  • Every bit of interest earned is based on the principal.
Imagine you have a garden. The principal amount is like the seed you plant. The seed is the foundation for the plant to grow over time.
Interest Rate
The interest rate is the percentage at which your investment grows over time. In the given exercise, the interest rate is \8\% per year. The interest rate, often denoted as \(\text{r}\), is typically expressed as a decimal when plugged into equations, so \8\% becomes 0.08. The interest rate affects how quickly your investment grows.

  • A higher interest rate means your investment grows faster.
  • A lower interest rate grows slower but is generally safer.

Think of the interest rate as the sunlight your plant needs to grow. The more sunlight (or higher interest rate), the faster the plant matures.
Compounding Periods
Compounding periods refer to how frequently the interest is added to the principal amount. In our exercise, the interest is compounded semiannually, meaning twice a year. The variable \(\text{n}\) denotes the number of compounding periods per year. For this exercise, \(\text{n} = 2\).

  • The more frequently interest is compounded, the more you earn.
  • Semiannual compounding means interest is added every six months.

Use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

In simpler terms, compounding periods are like how often you water your plant. More frequent watering (compounding periods) can lead to a more robust plant.

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Most popular questions from this chapter

Boxowitz, Inc., a computer firm, is planning to sell a new graphing calculator. For the first year, the fixed costs for setting up the new production line are 100,000 dollars. The variable costs for producing each calculator are estimated at 20 dollars. The sales department projects that 150,000 calculators can be sold during the first year at a price of 45 dollars each. a) Find and graph \(C(x),\) the total cost of producing \(x\) calculators. b) Using the same axes as in part (a), find and graph \(R(x),\) the total revenue from the sale of \(x\) calculators. c) Using the same axes as in part (a), find and graph \(P(x),\) the total profit from the production and sale of \(x\) calculators. d) What profit or loss will the firm realize if the expected sale of 150,000 calculators occurs? e) How many calculators must the firm sell in order to break even?

Determine the domain of each function. $$f(x)=\sqrt{5 x+4}$$

At most, how many \(y\) -intercepts can a function have? Explain.

The surface area of a person whose mass is 75 kg can be approximated by the function \(f(h)=0.144 h^{1 / 2}.\) where \(f(h)\) is measured in square meters and \(h\) is the person's height in centimeters. (Source: U.S. Oncology.) a) Find the approximate surface area of a person whose mass is 75 kg and whose height is 180cm. b) Find the approximate surface area of a person whose mass is 75 kg and whose height is 170cm. c) Graph the function \(f\) for \(0 \leq h \leq 200.\)

Radar range. The function given by \(R(x)=11.74 x^{0.25}.\) can be used to approximate the maximum range, \(R(x)\) in miles, of an ARSR-3 surveillance radar with a peak power of \(x\) watts. a) Determine the maximum radar range when the peak power is 40,000 watts, 50,000 watts, and 60,000 watts. b) Graph the function.
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