Problem 22 - Calculus Early Transcendentals

Question

A cyclist rides down a long straight road at a velocity (in \(\mathrm{m} / \mathrm{min}\) ) given by \(v(t)=400-20 t,\) for \(0 \leq t \leq 10\) where \(t\) is measured in minutes. a. How far does the cyclist travel in the first 5 min? b. How far does the cyclist travel in the first 10 min? c. How far has the cyclist traveled when her velocity is \(250 \mathrm{m} / \mathrm{min} ?\)

Step-by-Step Solution

Answer

Answer: The cyclist travels 1750 meters in the first 5 minutes, 3000 meters in the first 10 minutes, and 2437.5 meters when her velocity is 250 m/min.

Step 1: Find the Position Function

First, let's find the position function by integrating the velocity function. The position function is given as the integral of the velocity function with respect to time: $$s(t) = \int v(t) dt = \int (400 - 20t) dt$$ Let's integrate the function and find \(s(t)\).

Step 2: Integrate the Velocity Function

Integrating the velocity function with respect to \(t\) and adding an arbitrary constant \(C\), we have: $$s(t) = \int (400 - 20t) dt = 400t - 10t^2 + C$$ We are not given any initial position, so we can assume the cyclist starts at position \(0\) when \(t = 0\). Therefore, \(s(0) = 0\), and we can solve for \(C\): $$0 = 400(0) - 10(0)^2 + C \Rightarrow C = 0$$ So, the position function is: $$s(t) = 400t - 10t^2$$ Now we can use this function to find the required distances.

Step 3a: Distance Traveled in the First 5 Minutes

To find the distance traveled in the first 5 minutes, we need to find the position at \(t=5\) and then subtract the position at \(t=0\): $$s(5) - s(0) = (400(5) - 10(5)^2) - (400(0) - 10(0)^2) = 2000 - 250 = 1750\ \text{m}$$ So, the cyclist travels 1750 meters in the first 5 minutes.

Step 3b: Distance Traveled in the First 10 Minutes

Similarly, to find the distance traveled in the first 10 minutes, we need to find the position at \(t=10\) and then subtract the position at \(t=0\): $$s(10) - s(0) = (400(10) - 10(10)^2) - (400(0) - 10(0)^2) = 4000 - 1000 = 3000\ \text{m}$$ So, the cyclist travels 3000 meters in the first 10 minutes.

Step 3c: Distance Traveled When Velocity is 250 m/min

First, we need to find the time \(t\) at which the cyclist's velocity is $250\ \mathrm{m} / \mathrm{min}$: $$v(t) = 400 - 20t = 250 \Rightarrow t = \frac{(400 - 250)}{20} = \frac{150}{20} = 7.5$$ Now, we can find the distance traveled at \(t = 7.5\) minutes using the position function: $$s(7.5) - s(0) = (400(7.5) - 10(7.5)^2) - (400(0) - 10(0)^2) = 3000 - 562.5 = 2437.5\ \text{m}$$ So, the cyclist travels 2437.5 meters when her velocity is $250\ \mathrm{m} / \mathrm{min}$.

Key Concepts

Position Function

In calculus, understanding how different functions relate to real-world situations like motion is crucial. One of these functions is called the **position function**. The position function, often denoted as \(s(t)\), helps describe the location of an object over time. It is derived from the object's velocity function through the process of integration. This is because integration essentially "sums up" the velocities over a period of time to give a total change in position.

For our cyclist, the velocity function is given by \(v(t) = 400 - 20t\). Integrating this function helps us find the position function \(s(t)\). Let's do the integration step:

• The integral of the constant \(400\) with respect to \(t\) is \(400t\).
• For \(-20t\), the integral becomes \(-10t^2\).
• We also add an arbitrary constant, \(C\), to account for any initial position, but since \(s(0) = 0\), we find \(C = 0\).

Thus, the position function for our cyclist becomes:
\[ s(t) = 400t - 10t^2 \]
This function helps us understand exactly where the cyclist is at any given time \(t\), based on her initial velocity.

Velocity Function

The velocity function describes how fast an object's position changes with respect to time. In the context of our cyclist, the velocity function is \(v(t) = 400 - 20t\). This linear equation tells us that her velocity decreases as time \(t\) increases.

Here, let's break down what the velocity function reveals:

• Initially, when \(t = 0\), her velocity is \(400\) meters per minute. This shows a fast start.
• As time progresses, the term \(-20t\) means her speed decreases by \(20\) meters per minute for each minute that passes.
• Eventually, her velocity becomes zero when \(t = 20\) minutes because the linear equation decreases linearly with \(t\).

By identifying the velocity function and choosing specific values of \(t\), we can calculate how her speed changes and hence, determine when she slows down or stops riding.

Distance Traveled

Understanding the total distance traveled by an object over time involves evaluating the change in position, which is directly derived from its position function. For the cyclist, the task required us to find out how far she traveled over different time intervals by analyzing her position function \(s(t) = 400t - 10t^2\).

Let's see how this works in practice with some examples from our exercise:

• For the first 5 minutes, we calculated \(s(5)\) to find the distance she traveled. By inputting \(t = 5\) into the position function, we get \(1750\) meters.
• Next, we assessed \(s(10)\) to determine the distance in the first 10 minutes, resulting in \(3000\) meters.
• We also looked at a scenario where her velocity was \(250\) meters per minute to find \(t = 7.5\). Substituting into the position function, we got \(2437.5\) meters.

These calculations offer concrete answers to the cyclist's journey, illustrating how integral calculus can bridge the gap between velocity and actual ground covered.