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

After 1 hour: 1 meter, 2 hours: 2 meters, 3 hours: 3 meters. The distance function is \(d(t) = t\), a polynomial of degree 1.

Step 1: Understanding the Problem

We have a beetle walking in a straight line at a constant speed of 1 meter per hour. We need to find the distance covered after 1 hour, 2 hours, and 3 hours, and write an equation for distance as a function of time.

Step 1: Distance After 1 Hour

To find the distance covered after 1 hour, multiply the speed by the time: \(1\ \text{m/h} \times 1\ \text{h} = 1\ \text{meter}\).

Step 2: Distance After 2 Hours

To find the distance covered after 2 hours, multiply the speed by the time: \(1\ \text{m/h} \times 2\ \text{h} = 2\ \text{meters}\).

Step 3: Distance After 3 Hours

To find the distance covered after 3 hours, multiply the speed by the time: \(1\ \text{m/h} \times 3\ \text{h} = 3\ \text{meters}\).

Step 4: Writing the Distance Function

The distance covered \(d\), as a function of time \(t\), is given by the equation \(d(t) = 1 \times t\). In simplified form, the equation is \(d(t) = t\).

Step 5: Verifying the Degree of the Polynomial

The function \(d(t) = t\) can be rewritten as \(d(t) = 1t^1 + 0\), which shows that this is a polynomial of degree 1, as the highest power of \(t\) is 1.

Key Concepts

Polynomial Degree

A polynomial is an expression made up of terms, each term being a product of a constant and a variable raised to a non-negative integer power. The degree of a polynomial is one of its fundamental characteristics. It represents the highest power of the variable in the polynomial. The degree gives us insight into the behavior and shape of the graph that represents the polynomial.

In the exercise provided, the beetle's walking function is expressed as \( d(t) = t \). We can also express this as \( d(t) = 1t^1 + 0 \). The term \( 1t^1 \) shows the highest power of the variable \( t \), which is 1. Therefore, this function is a polynomial of degree 1. The coefficient "1" is the constant multiplier of \( t \) and the zero term does not change the degree. A polynomial of degree 1 is also known as a linear function because its graph is a straight line.

Distance-Time Relationship

The distance-time relationship is a way to describe how the distance covered changes over time. It is often represented by a mathematical function. In the context of the given problem, our beetle travels at a linear rate.

In time-related movement problems, the equation used is usually in the form of \( d = rt \) (distance \( = \) rate \( \times \) time). Here, the rate (or speed) is constant, which makes the relationship special. This particular setup allows us to predict the distance traveled at any given time, considering the speed never changes. Our equation becomes \( d(t) = t \), meaning the distance is directly proportional to the time when the speed is constant. For each hour the beetle walks, it covers an additional meter, showing a perfectly linear relationship.

Constant Speed

Constant speed refers to a scenario where the speed of an object does not change over time. This means that the object moves the same distance every unit of time. It's a vital concept for understanding and predicting an object's movement over time.

In this exercise, the beetle moves at 1 meter per hour, a constant speed. Because the speed is constant, the formula for the distance becomes straightforward: simply multiply this constant speed by the number of hours walked. This results in a simple mathematical model where you can determine how far the beetle travels after any number of hours just by multiplying the time by the speed. Constant speed implies predictability, allowing us to use simple linear functions, like \( d(t) = t \), to describe motion over time. This simplicity makes analyzing the beetle's pathway easy and straightforward.