Question

Use a graphing utility to solve each equation for $x.$ $$25=50 e^{-0.05 x}$$

   Use a graphing utility to solve each equation for $x.$
$$25=50 e^{-0.05 x}$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 4, Problem 58

Instant Answer

Solved by Expert Charles Carter

07/01/2020

Step 1

This gives us: $$\frac{25}{50} = e^{-0.05x}$$ which simplifies to: $$0.5 = e^{-0.05x}$$ Show more…

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Use a graphing utility to solve each equation for $x.$ $$25=50 e^{-0.05 x}$$

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Key Concepts

Exponential Functions

Exponential functions are expressions where a constant base is raised to a variable exponent, often modeled as f(x) = a e^(bx). These functions describe processes that grow or decay at rates proportional to their current value. In solving equations, understanding the behavior of exponential functions is crucial, as these functions are invertible, typically using logarithms.

Graphical Methods for Solving Equations

Graphical methods involve plotting functions on a coordinate system and identifying the points where they intersect. This approach is particularly useful when equations are difficult to solve algebraically. By graphing both sides of an equation, one can visually determine the solution as the x-coordinate of the intersection point.

Intersection of Functions

The intersection of functions is a key concept wherein the solution to an equation is found by identifying the point(s) where two graphs meet. When each side of an equation is treated as a separate function, their intersection represents the values that satisfy the equality, which is often determined using graphing utilities.

Exponential Decay

Exponential decay is a specific type of exponential function where the exponential component decreases over time, typically characterized by a negative exponent. This concept is important in various fields such as physics and finance, as it accurately models processes that diminish over time. Understanding decay behavior aids in setting up and solving related equations graphically or algebraically.

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