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Chapter 4: Problem 81

Solve each equation. Round answers to four decimal places. $$(1+r)^{3}=2.3$$

Short Answer

Expert verified
r 鈮� 0.3070

Step by step solution

01

Understand the Equation

The given equation is \(1+r)^{3}=2.3\). It states that \(1+r\) raised to the power of 3 equals 2.3.
02

Isolate the Expression

To isolate \(1+r\), take the cube root of both sides. This gives us \(1+r = (2.3)^{\frac{1}{3}}\).
03

Calculate the Cube Root

Using a calculator, find \((2.3)^{\frac{1}{3}}\), which approximates to 1.3070 (rounded to four decimal places).
04

Solve for r

Subtract 1 from both sides to solve for \r\: \(r = 1.3070 - 1\).
05

Final Calculation

Perform the subtraction: \(r = 0.3070 \), rounded to four decimal places.

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Key Concepts

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

Exponential Equations
Exponential equations involve variables in the exponent. In our problem, the equation was \( (1+r)^{3}=2.3 \). This is a classic example of an exponential equation.
To solve these equations, we need to use specific methods to isolate the variables.
Here are some key steps:
  • Identify the equation and the variable in the exponent.
  • Apply an operation (like taking the cube root) to both sides to simplify it.
Understanding exponential equations is crucial for precalculus, as they come up often in various mathematical contexts.
Cube Root
A cube root is the number that, when multiplied by itself three times, equals the given number. In this exercise, to isolate \(1+r\), we need to take the cube root of both sides of the equation: \( 1+r = (2.3)^{\frac{1}{3}} \).
This operation helps us simplify and solve the equation.
  • Cube roots are the inverse of cubing a number.
  • We often use calculators to find cube roots for non-perfect cubes.
Taking the cube root is an essential skill in solving higher-degree polynomial equations.
Rounding
Rounding numbers makes them simpler and easier to work with, especially when dealing with decimals. In the given exercise, the final step involved rounding \(1.3070\) to four decimal places.
  • Identify the number of decimal places required.
  • Look at the next digit to decide whether to round up or keep the same value.
Properly rounding results is crucial in mathematics to ensure accuracy without unnecessary complexity.

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Most popular questions from this chapter

Solve each problem. In Nigeria, deforestation occurs at the rate of about \(5.2 \%\) per year. Assume that the amount of forest remaining is determined by the function $$F=F_{0} e^{-0.052 t},$$ where \(F_{0}\) is the present acreage of forest land and \(t\) is the time in years from the present. In how many years will there be only \(60 \%\) of the present acreage remaining?

To evaluate an exponential or logarithmic function we simply press a button on a calculator. But what does the calculator do to find the answer? The next exercises show formulas from calculus that are used to evaluate \(e^{x}\) and \(\ln (1+x)\). Infinite Series for Logarithms The following formula from calculus can be used to compute values of natural logarithms: $$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots$$ where \(-1< x <1 .\) The more terms that we use from the formula, the closer we get to the true value of \(\ln (1+x)\) Find \(\ln (1.4)\) by using the first five terms of the series and compare your result to the calculator value for \(\ln (1.4)\)

Solve each problem. When needed, use 365 days per year and 30 days per month. National Debt The national debt was about \(\$ 10\) trillion in 2008 - a. If the United States paid \(5.5 \%\) interest compounded continuously on the debt, then what amount of interest does the government pay in one day? b. How much is saved in one day if the interest were \(5.5 \%\) compounded daily?

Work in a small group to write a summary (including drawings) of the types of graphs that can be obtained for logarithmic functions of the form \(y=\log _{a}(x)\) for \(a>0\) and \(a \neq 1\).

Find the approximate solution to each equation. Round to four decimal places. $$10^{x}-3=5$$
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