Problem 85 - College Algebra

Question

The function \(P(t)=145 e^{-0.022 t}\) 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. \([\mathrm{TRACE}]\) 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

Based on this solution method, the runner's pulse will drop to 70 beats per minute approximately 16.8 minutes after the race. This observation aligns with the graphical observation and confirms the validity of both methods of solution.

Step 1: Graph the Function

Use a graphing utility to plot the function, marking the \(y\)-coordinate as the pulse and the \(x\)-coordinate as time. The function given is an exponential decay function \(P(t)=145 e^{-0.022 t}\). Identify the pulse rate of 70 on the \(y-\) axis.

Step 2: Find the Intersection Point

Activate the trace mode on the graphing utility and move along the curve of the function until the y-value reads close to 70. The corresponding \(x\)-value is the approximate minutes after the race when the runner's pulse drops to 70 beats per minute.

Step 3: Solve the Equation Algebraically

To confirm the result obtained graphically, set \( P(t) = 70 \) in the function \( P(t) = 145 e^{-0.022 t} \) and solve for \( t \). This way, you get the equation \(70 = 145 e^{-0.022 t}\). Divide both sides of the equation by 145, yielding \( \frac{70}{145} = e^{-0.022 t} \). Taking the natural logarithm of both sides will then help to isolate \( t \).

Step 4: Calculate t

To isolate \(t\), use the property of logarithms \( ln e^{x} = x ln e \), thus we get \(-0.022 t = ln \left( \frac{70}{145}\right)\) . You can calculate \(t\) by dividing both sides by \(-0.022 .\) Round the result to the nearest tenth.

Key Concepts

Graphing Functions

Graphing functions helps us visualize how variables change in relation to each other. In this exercise, we graph the function \(P(t)=145 e^{-0.022 t}\), which represents a runner's pulse over time after a race. To graph it, you need: an understanding of axes, where the \(x\)-axis represents time \(t\) (in minutes), and the \(y\)-axis shows the pulse rate \(P(t)\). For exponential functions like ours:

• The graph typically decreases if the exponent is negative, signaling a decay.
• The y-intercept starts at the initial pulse, \(P(0)=145\).

Using a calculator or plotter, you input the equation and visually trace the slope downward, looking for changes like the pulse rate reaching 70 beats per minute. This task involves ensuring the graph's scale accurately depicts time (\(0 \leq t \leq 15\)) and closely follows the decay pattern.

Solving Equations Algebraically

Solving equations algebraically involves using mathematical operations to find unknown values. Here, we want to find when the runner's pulse is 70 beats per minute. Start with the equation \(70 = 145 e^{-0.022 t}\) to determine \(t\):

• First, divide both sides by 145 to isolate the exponential term: \( \frac{70}{145} = e^{-0.022 t}\).
• Next, apply the natural logarithm (\(\ln\)) to both sides to deal with the exponential nature, since \(\ln(e^x) = x\).

This forms \(-0.022 t = \ln(\frac{70}{145})\), allowing further steps to solve for \(t\) and confirm the approximate minutes post-race that the pulse decreases to 70. Algebraic solutions verify graphical insights, ensuring precision and reinforcing your understanding of both graphical and algebraic methods.

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent. They show different behaviors based on the nature of the exponent:

• If positive, they grow rapidly.
• If negative, as in \(P(t)=145 e^{-0.022 t}\), they decay over time.

In our scenario, \(e^{-0.022 t}\) captures an exponential decay, indicating a decrease in the runner's pulse over time. Understanding exponential decay is critical as it appears in numerous applications:

• Radioactive decay.
• Cooling of hot objects.
• Population declines or even finance scenarios.

Each scenario shares the underlying concept that the decay process slows as time progresses, which is visually apparent in graphs of these functions becoming less steep.

Natural Logarithms

Natural logarithms are the inverse of exponential functions based on the constant \(e\) (approximately 2.718). This special logarithm, represented as \(\ln\), helps in simplifying equations involving \(e\) by canceling the exponential:

• The identity \(\ln(e^x) = x\) is crucial for solving equations algebraically.
• By converting exponential forms into logarithmic form, calculations simplify, allowing for straightforward isolation of variables.

In our problem, using \(\ln\) allowed us to tackle the exponentially decaying pulse function, leading effectively to solvable linear equations. Logs are invaluable for working through equations with both exponential growth and decay, as showcased when balancing chemical equations, assessing complex financial models, or converting different scaling units.