Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$g(t)=e^{0.09 t}-3$$

Step-by-Step Solution

Answer

The zero of the function \(g(t)=e^{0.09 t}-3\) is approximately at t-value which can be found using a graphing utility and where the graph intersects the t-axis and approximation should be to three decimal places.

Step 1: Understand the Function

First, understand the given function \(g(t)=e^{0.09 t}-3\). This is an exponential function where base e (the natural logarithm base) is raised to the power of a linear function 0.09t and the result is subtracted by 3.

Step 2: Graph the Function

Next step is to graph the function using a graphing utility. Feed the function \(g(t)=e^{0.09 t}-3\) into the graphing software. Plot the function over a reasonable range that allows you to see where the function crosses the t-axis.

Step 3: Identify Zero of the Function

Observe the point on the t-axis where the graph intersects. The zero or root of an equation is the value that makes the equation equals to zero. Approximate this value t accurate to three decimal places by using the cursor or the trace function on the graphing utility.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions. For the function \(g(t) = e^{0.09t} - 3\), they allow us to quickly find where it crosses the horizontal axis, an important aspect in identifying roots. To start, input the function into any graphing utility. Common graphing utilities include handheld calculators, online graphing calculators, or even graphing software like Desmos.

Once you graph \(g(t)\), adjust the viewing window to ensure the graph displays a clear intersection with the t-axis. This intersection is crucial because it's where the function equals zero, known as its root.

By having a visual representation, graphing utilities provide insight into the behavior of the function, such as growth patterns dictated by the exponential term \(e^{0.09t}\), and how the graph shifts down due to the subtraction of 3.

Roots of Equations

Roots of equations are the values of the variable that make the equation equal to zero. For \(g(t) = e^{0.09t} - 3\), we want to find the value of \(t\) which makes \(g(t) = 0\). This is also called a 'zero'. Identifying these roots is essential in understanding where and how a function balances out to zero.

Mathematically, finding the root means solving \(e^{0.09t} - 3 = 0\), which simplifies to \(e^{0.09t} = 3\). This equation can be solved by using logarithms, specifically the natural logarithm, to isolate \(t\). However, graphical methods often provide a more intuitive understanding and can be quickly done with modern technology.

Approximate Solutions

Often, graphing utilities give us an approximate value for the roots. This approximation is usually accurate to a specified number of decimal places, such as three in this exercise. Once you use a graphing utility to find where \(g(t)\) crosses the t-axis, it is essential to zoom in or use the trace function provided by the utility to precisely identify the root.

Since the root is rarely an integer, the approximation provides a practical answer to equations that might be difficult or impossible to solve analytically.

• Evaluate the function close to the zero using the graph's tool.
• Use the cursor to hover over the intersection point.
• Utilize the trace or zoom feature to read the root's precise value.

This method improves efficiency and accuracy when evaluating complex equations, especially those involving non-linear or exponential functions.