Question

You buy a new stereo for \(\$ 1300\) and are able to sell it 4 years later for \$275. Assume that the resale value of the stereo decays exponentially with time. Write an equation giving the resale value \(V\) (in dollars) of the stereo as a function of the time \(t\) (in years) since you bought it.

Short answer

The resale value of the stereo as a function of the time since it was bought is given by the exponential decay equation \(V = 1300 * e^{-rt}\), where t is time and r is the decay rate which is \(-ln(275/1300) / 4\).

Step-by-step solution

Identify given values

\$1300 is the initial value (P) and \$275 is the value (V) after 4 years (t).

Write down the exponential decay formula

Using the exponential decay formula \(V = P * e^{-rt}\), plug in the given values. This gives \(275 = 1300 * e^{-4r}\).

Solve for r

To solve for r, rearrange the equation to isolate r. This gives \(e^{-4r} = 275/1300\). Then take the natural logarithm of both sides. This gives \(-4r= ln(275/1300)\), then \(r = -ln(275/1300) / 4\).

Plug the value of r back into the equation

Having obtained the decay rate r, plug this value back into the initial decay equation to secure the resale value as a function of time. This gives: \(V = 1300 * e^{-rt}\)

Key concepts

Exponential Decay Formula

Understanding the exponential decay formula is crucial for solving problems that deal with decreasing values over time, such as the depreciation of assets or the decay of radioactive substances. The general exponential decay formula can be represented as:

\[ V = Pe^{-rt} \]
where \( V \) is the final amount remaining after time \( t \) has passed, \( P \) is the initial amount, \( r \) is the decay rate, and \( e \) is the base of natural logarithms, approximately equal to 2.71828. The negative sign in the exponent indicates that the quantity is decreasing over time.

In the context of our problem, you start with a stereo worth \$1300 \(P\), which decays to a value of \$275 \(V\) after 4 years \(t\). Using these values with our formula can determine the decay rate \(r\) and subsequently predict the stereo's future value at any given time. It's essential to get comfortable with this equation, as it's commonly used in various fields, from finance to physics.

Solving Exponential Equations

Solving exponential equations such as those found in exponential decay problems involves isolating the variable you're solving for, usually the decay rate \( r \) or the time \( t \). The strategy often involves taking the natural logarithm of both sides of the equation because the natural logarithm function is the inverse of the exponential function, simplifying the equation and making it possible to solve for the variable.

In our stereo problem, you found the equation \( 275 = 1300 * e^{-4r} \). To solve for \( r \) we first divide both sides by 1300, leading to \( e^{-4r} = 275/1300 \). We then apply the natural logarithm to both sides, resulting in \( -4r = ln(275/1300) \). Dividing by -4 gives us the decay rate \( r \).

This process highlights a key concept: logarithms are used to 'unpack' the exponent, allowing us to solve for variables that are otherwise hard to isolate. It's this move from exponential to logarithmic form that is crucial in solving many types of exponential equations.

Natural Logarithms

Natural logarithms are a fundamental part of solving exponential decay problems and are denoted as \( ln \). The natural logarithm is the logarithm with base \( e \) where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It's called 'natural' because it arises naturally in mathematics when dealing with continuous growth or decay processes.

The property that \( ln(e^x) = x \) is what makes \( ln \) so valuable for solving equations with \( e \) as their base. For instance, if we take the natural logarithm of both sides of the equation \( e^x = y \), we get \( ln(e^x) = ln(y) \) and, using the property mentioned above, \( x = ln(y) \).

Applying this to our stereo example, by taking the natural logarithm of both sides of our decay equation, we simplified the problem and found the decay rate \( r \). Understanding natural logarithms and their relationship with exponential functions widens your mathematical toolkit, enabling the solution of a wide range of algebraic problems.