Problem 62 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, solve for \(x\) or \(t\). $$ 400 e^{-0.0174 t}=1000 $$

Step-by-Step Solution

Answer

By following the outlined steps, \(t = \frac{ln(2.5)}{-0.0174}\).

Step 1: Divide by 400 to isolate the exponential function

Divide both sides of the equation by 400 to isolate the exponential on one side. This results in the equation \(e^{-0.0174 t} = \frac{1000}{400} = 2.5\).

Step 2: Take natural logarithm of both sides

To eliminate the exponent on \(t\), apply natural logarithm (ln) to both sides of the equation. The rule of logarithm used here is \(ln(e^{x}) = x\). This gives us the equation \(-0.0174 t = ln(2.5)\).

Step 3: Solve for \(t\)

To obtain the value of \(t\), divide both sides of the equation by -0.0174. So, \(t = \frac{ln(2.5)}{-0.0174}\).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \text{ln}, is a mathematical function that’s the inverse of the exponential function with a base of Euler's number, approximately equal to 2.718281. This special base is what gives the natural logarithm its name. When you have an equation like \( e^x = y \), you can apply the natural logarithm to both sides to isolate the exponent, such that \( \text{ln}(e^x) = \text{ln}(y) \), which simplifies to \( x = \text{ln}(y) \). This property makes it a powerful tool when solving exponential equations.

Isolate the Variable

To solve an equation for a specific variable means to isolate that variable on one side of the equation. In the given exercise, the goal is to isolate \( t \). The process typically involves performing the same operation on both sides of the equation to maintain equality while systematically getting the variable alone. In our example, the first step was to divide by 400, which resulted in an easier equation to work with, containing the isolated exponential term. Only after this step could the natural logarithm be effectively applied to both sides, moving us closer to isolating \( t \) completely. It's a sequence of strategic moves, like a dance that leads you to the value of the variable you seek.

Exponential Decay

Exponential decay describes the process of a quantity decreasing over time at a rate proportional to its current value. It's often expressed by the formula \( P(t) = P_0 e^{-kt} \), where \( P(t) \) is the amount at time \( t \), \( P_0 \) is the initial amount, \( e \) is Euler's number, and \( k \) is the decay constant. In this exercise, the quantity is decaying from 400 to 1000 (which indicates that there might be a mistake, usually, decay would mean it's decreasing), and the negative sign in the exponent \( -0.0174t \) signifies a decrease over time. By solving the equation, we find the specific point in time when the quantity has decayed to a certain level. This concept is widely seen in real-life scenarios such as radioactive decay, cooling of hot substances, and depreciation of asset values.