Problem 49 - Calculus for Biology and Medicine

Question

Prof. Roper drives to work in stop-and-go traffic. His speed measured in miles per hour (mph) is given by the following function of time, \(t\), measured in minutes $$ v(t)=30+20 \sin (t / 5) $$ The total journey time is 1 hour. Explain using the MVT why the total distance that he travels in this hour is somewhere between 10 and 50 miles.

Step-by-Step Solution

Answer

The distance traveled is between 10 and 50 miles due to speed bounds (10-50 mph) by sine oscillations and MVT.

Step 1: Define the Problem Context

Prof. Roper's speed during his 1-hour journey is given by the function \( v(t) = 30 + 20 \sin(t/5) \), where \( t \) is in minutes and \( v(t) \) in miles per hour. We aim to find out why the distance traveled lies between 10 and 50 miles using the Mean Value Theorem (MVT).

Step 2: Understanding the Mean Value Theorem (MVT)

The MVT states that for a continuous function on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \( c \) in \((a, b)\) where the instantaneous rate of change \( f'(c) \) is equal to the average rate of change over \([a, b]\).

Step 3: Calculate the Average Speed

To use MVT, calculate the average speed over 1 hour (60 minutes):\[\text{Average speed} = \frac{1}{60} \int_0^{60} v(t) \, dt\]We approximate this by noting that the average of \( v(t) \) depends on evaluating the integral of \( v(t) \). Let's find this bound briefly by considering the nature of \( v(t) \).

Step 4: Bound the Function

Analyze \( v(t) = 30 + 20 \sin(t/5) \). The sine function, \( \sin(t/5) \), oscillates between -1 and 1:- The minimum value of \( v(t) \) is when \( 20 \sin(t/5) = -20 \), then \( v(t) = 10 \). - The maximum value is when \( 20 \sin(t/5) = 20 \), giving \( v(t) = 50 \).

Step 5: Calculate Distance Bounds

Since the speed function \( v(t) \) varies from 10 to 50 mph, the minimum possible average speed is 10 mph and the maximum is 50 mph over the 1 hour (60 minutes) journey. Thus, the total distance traveled lies between:\[60 \times 10 = 10 \text{ miles} \text{ (minimum) and } 60 \times 50 = 50 \text{ miles} \text{ (maximum)}.\]

Step 6: Apply MVT Conclusion

By MVT, there exists at least one time \( c \) during the journey where \( v(c) \) equals the average speed over the interval. Given the bounds derived, the distance traveled, which equates to the integral of \( v(t) \) over 60 mins, will also lie between 10 and 50 miles.

Key Concepts

Integral Calculus

Integral calculus is all about finding the accumulation of quantities.
When dealing with motion, such as Prof. Roper's journey, integral calculus helps to calculate the total distance traveled over a period of time based on his speed function.

The speed function given is:

• \( v(t) = 30 + 20 \sin(t/5) \)

This function shows speed at every moment over his 1-hour trip, where time \( t \) is in minutes.
To find the total distance, we integrate this function over the time interval:

• \( \int_0^{60} v(t) \ dt \)

Essentially, this integral sums all the small distance increments over time, giving the entire journey's distance.
The result of this calculation is the precise distance, based on the curve of his speed function.

Distance Calculation

Distance calculation in the context of Prof. Roper's trip involves determining how far he travels based on his speed at any given time.
Let's break down how this is done using the Mean Value Theorem (MVT).

The speed function, \( v(t) = 30 + 20 \sin(t/5) \), helps us to determine how fast he's moving at any minute of his journey.
The total journey takes place over an hour, or 60 minutes.

• To calculate the minimum distance he travels, we consider the lowest speed, which is 10 mph. Over 60 minutes, he'd cover:\( 60 \times 10 = 10 \) miles.
• For the maximum distance, we consider the highest speed, 50 mph, resulting in:\( 60 \times 50 = 50 \) miles.

This establishes the bounds for the total distance: somewhere between 10 and 50 miles.
The actual distance lies within these bounds because the speed fluctuates between these limits during the journey.

Trigonometric Functions

Trigonometric functions, like sine, play a key role in dictating how Prof. Roper's speed changes over time.
The function \( v(t) = 30 + 20 \sin(t/5) \) incorporates a sine component, which causes cyclical changes in speed.

Here's a closer look:

• The sine function, \( \sin(t/5) \), oscillates between -1 and 1.
  This means it periodically affects the speed with values between 10 mph (minimum) and 50 mph (maximum).
• The periodic nature of sine means these peaks and troughs repeat within the time frame, introducing a regular pattern to the speed changes.

When used in motion problems like this, sine smoothens out abrupt changes, providing a realistic model of speed variation over time.
This explains why the speed isn't constant, leading to the varied range of possible distances within the hour.