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Chapter 3: Problem 77

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$e^{0.09 t}=3$$

Short Answer

Expert verified
The graphical solution would be around \( t \approx 10.416 \). The confirmed algebraic solution is \( t = \frac{ln(3)}{0.09} \approx 10.416 \). Verification is done by substitifying the obtained value back into the original equation and making sure that it holds.

Step by step solution

01

Graph the Function

Use a graphing utility to graph the equation \( e^{0.09 t} = 3 \). The graph of this equation would show where the function intersects the line \( y = 3 \), the points of intersection represent the solution of the equation.
02

Approximate the Solution

By looking at the graph, observe the point(s) at which the function crosses the line \( y = 3 \). Approximate the value of \( t \) at this point up to three decimal places.
03

Algebraically Solve the Equation

To verify the solution, solve the equation algebraically. To do so, apply the natural logarithm (ln) to both sides of the equation, resulting in \( 0.09 t = ln(3) \). Then, solve for \( t \) by dividing both sides by 0.09, and compare this result with the approximation obtained from the graph.

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