Problem 135 - Algebra and Trigonometry

Question

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Step-by-Step Solution

Answer

The runner's pulse will be 70 beats per minute after approximately 8.6 minutes.

Step 1: Graphing the Function

Plot the function \(P(t)=145 e^{-0.092 t}\) on a graphing utility for the domain \(0 \leq t \leq 15 \). TRACE along the curve to observe when the pulse rate drops to 70 beats per minute. This visualization will give a preliminary understanding of when the pulse rate drops to 70, which we will then confirm algebraically. Note down the approximate 't' value.

Step 2: Setting up the Equation

In order to find the exact time when the runner's pulse is 70 beats per minute, set the pulse \(P(t)\) at 70 and solve for \(t\). You can do this by setting up the following equation: \(70 = 145 e^{-0.092 t}\)

Step 3: Solving the Equation

Solving the equation involves isolating \( t \). After rearranging, we'll get a new equation: \(-0.092 t = \ln(\frac{70}{145})\). Dividing both sides by \(-0.092\) gives \(t = - \frac{\ln(\frac{70}{145})}{0.092}\). Use a calculator to find the value of t. Round your answer to the nearest tenth of a minute.