Problem 31 - Calculus for Biology and Medicine

Question

Suppose that a beetle walks up a tree along a straight line at a constant speed of 1 meter per hour. What distance will the beetle have covered after 1 hour, 2 hours, and 3 hours? Write an equation that expresses the distance (in meters) as a function of the time (in hours), and show that this function is a polynomial of degree 1 .

Step-by-Step Solution

Answer

The beetle covers 1, 2, and 3 meters in 1, 2, and 3 hours, respectively. The function is \( d(t) = t \), a polynomial of degree 1.

Step 1: Understanding the Relationship

Since the beetle walks at a constant speed of 1 meter per hour, the distance covered by the beetle is directly proportional to the time it has walked. This means the relationship between the distance and time is linear.

Step 2: Writing the Function

The distance covered by the beetle (\( d \)) in meters is a function of time (\( t \)), where the speed is 1 meter per hour. This relationship can be expressed as \( d = 1 \cdot t \). Here, the distance (\( d \)) is equal to the speed (1 meter/hour) times the time (\( t \)), which simplifies to \( d = t \).

Step 3: Evaluating the Distance at Given Times

Using our function \( d = t \):- After 1 hour (\( t = 1 \)): \( d = 1 \).- After 2 hours (\( t = 2 \)): \( d = 2 \).- After 3 hours (\( t = 3 \)): \( d = 3 \).So, the beetle covers 1 meter after 1 hour, 2 meters after 2 hours, and 3 meters after 3 hours.

Step 4: Identifying the Polynomial Degree

The function \( d = t \) is a polynomial. It can be written as \( d(t) = 1t + 0 \), which is of the form \( ax + b \), where \( a = 1 \) and \( b = 0 \). This confirms it is a polynomial of degree 1, since the highest exponent of \( t \) is 1.

Key Concepts

Linear Relationship

When we talk about a linear relationship, we're describing a situation where two quantities have a direct, straight-line relationship with each other. This means that as one value increases or decreases, the other does so in a predictable way. In the case of our beetle, it walks up the tree at a constant speed, so the distance it covers over time forms a straight line when plotted on a graph. The simplicity of a linear relationship makes it easier to predict future values. For the beetle, its distance from the starting point always changes the same amount for each hour that passes. If you were to draw this on a graph, you'd notice that the line doesn't bend or curve; it stays straight. The formula associated with a linear relationship is often in the form of \( y = mx + c \), where \( m \) is the slope (or rate of change) and \( c \) is the initial value of \( y \). In this exercise, the formula is simplified to \( d = t \), indicating a direct, unchanging rate.

Constant Speed

Constant speed refers to a condition where a moving object, like our beetle, maintains the same speed over time without speeding up or slowing down. This consistency is essential for determining the linear relationship between time and distance. For our story of the beetle, it moves at 1 meter per hour without variance. This steady pace makes calculations straightforward because each unit of time correlates to an identical addition to the total distance traveled.

• A constant speed simplifies the process of calculating distance over time. Simply multiply the speed by the time elapsed to find the distance.
• Conversely, if you know how far the beetle has traveled, you can determine how long it has been moving simply by dividing the total distance by the speed.
• In situations like the beetle's, where the speed is steady, any changes in distance over time will reflect that constant speed exactly.

This concept is crucial in various real-world applications, not only when considering tiny beetles but also in larger scales like cars moving along highways.

Distance-Time Function

The distance-time function establishes a model for computing how far an object travels over time. In this exercise, it is portrayed with the polynomial expression \( d = t \). Here, \( d \) represents the distance the beetle travels, while \( t \) is the time elapsed. This function is a polynomial of degree 1, indicating it has a straightforward linear form. The relationship here is simple: time increases by one unit, and distance increases by one unit of measurement. Such a simple polynomial, also known as a linear function, holds unique properties.

• The slope of our distance-time function equals the beetle's speed, which is 1 meter per hour in this case.
• This makes the graph of the function a straight line, consistent with its linear nature.
• Polynomials of degree 1, like this one, are unique in their predictability and straightforward calculation methods.

This function is a classic example of how mathematical models can help simplify and predict movement and other changes over time effortlessly.