Problem 56 - Calculus Early Transcendentals

Question

At noon \((t=0),\) Alicia starts running along a long straight road at \(4 \mathrm{mi} / \mathrm{hr}\). Her velocity decreases according to the function \(v(t)=4 /(t+1),\) for \(t \geq 0 .\) At noon, Boris also starts running along the same road with a 2 -mi head start on Alicia; his velocity is given by \(u(t)=2 /(t+1),\) for \(t \geq 0\). a. Find the position functions for Alicia and Boris, where \(s=0\) corresponds to Alicia's starting point. b. When, if ever, does Alicia overtake Boris?

Step-by-Step Solution

Answer

Answer: Alicia overtakes Boris at approximately \(t = 1.72\) hours.

Step 1: Find the position function for Alicia

To find the position function for Alicia, we integrate her velocity function. Alicia's velocity function is given as \(v(t) = \frac{4}{t + 1}\). We will integrate it w.r.t time, \(t\): \(s_A(t) = \int v(t) dt = \int \frac{4}{t+1} dt\) We can solve this integral using a simple u-substitution, let's use \(u=t+1\), so \(du=dt\). Now the integral becomes: \(s_A(t) = \int \frac{4}{u} du = 4 \int \frac{1}{u} du\) Now we can integrate: \(s_A(t) = 4 \ln|u| + C = 4\ln|(t+1)| + C\) Since Alicia starts at position \(s=0\) at time \(t=0\), we can find the value of the constant \(C\): \(0 = 4 \ln|(0+1)| + C \Rightarrow C=0\) Therefore, Alicia's position function is: \(s_A(t) = 4\ln|(t+1)|\)

Step 2: Find the position function for Boris

Similarly, to find the position function for Boris, we need to integrate his velocity function too: \(u(t) = \frac{2}{t + 1}\) \(s_B(t) = \int u(t) dt = \int \frac{2}{t+1} dt\) Using the same u-substitution as before, let \(u=t+1\), we get: \(s_B(t) = \int \frac{2}{u} du = 2 \int \frac{1}{u} du\) After integrating, we get: \(s_B(t) = 2 \ln|u| + C = 2\ln|(t+1)| + C\) At \(t=0\), Boris starts 2 miles ahead of Alicia's starting point, so when \(s=0\), \(s_B(0)=2\): \(2 = 2\ln|(0+1)| + C \Rightarrow C=2\) Thus, Boris' position function is: \(s_B(t) = 2\ln|(t+1)| + 2\)

Step 3: Determine when Alicia overtakes Boris

Alicia overtakes Boris when their position functions are equal: \(s_A(t) = s_B(t)\) So, we have the equation: \(4\ln|(t+1)| = 2\ln|(t+1)| + 2\) To solve this equation, subtract \(2\ln|(t+1)|\) from both sides: \(2\ln|(t+1)| = 2\) Now, divide both sides by 2: \(\ln|(t+1)| = 1\) Take the exponential of both sides: \(t+1 = e^1\) Now, subtract 1 from both sides to obtain the time when Alicia overtakes Boris: \(t = e^1 - 1 \approx 1.72\) So, Alicia overtakes Boris at approximately \(t = 1.72\) hours.

Key Concepts

Position Function

When analyzing motion, understanding position functions is crucial. These functions describe the location of an object along a path at a given time. For Alicia and Boris, we determine their position by integrating their velocity functions with respect to time.
For Alicia, her position function is found by integrating her velocity function, \(v(t) = \frac{4}{t+1}\). After integrating using u-substitution and determining the constant of integration, we establish Alicia's position function as:

• \(s_A(t) = 4\ln|(t+1)|\)

For Boris, starting with a head start, his velocity function is \(u(t) = \frac{2}{t+1}\). Following the same integration process and accounting for his initial position, Boris's position function becomes:

• \(s_B(t) = 2\ln|(t+1)| + 2\)

Position functions help us visualize and mathematically determine the path and distance covered by each runner, setting ground for comparisons and calculations like who overtakes whom.

Velocity Function

Velocity functions outline the speed of an object and indicate its direction over time. Velocity is the derivative of the position function and provides insights into how fast the position changes.
For Alicia, the velocity function is \(v(t) = \frac{4}{t+1}\), demonstrating a speed that decreases as time progresses. This decline happens because the denominator in the function increases continuously, reducing the overall fraction value.
Similarly, Boris's velocity is \(u(t) = \frac{2}{t+1}\), which also decreases over time but starts at a slower rate compared to Alicia.
Understanding velocity functions helps us not only compute acceleration and positions but also figure out relative speeds and moments where one runner might overtake another.

U-substitution

U-substitution is a handy technique used in calculus to simplify the integration process. It involves substituting a complicated component of the integral with a single variable, making the integral easier to evaluate.
In the case of the given problem, we applied u-substitution to find the position functions of both Alicia and Boris.

• We set \(u=t+1\), simplifying integration of fractions to \( \frac{1}{u} \).
• This shifted the focus from variables with denominators to natural logarithm forms, simplifying the process.

Integrating using u-substitution made handling natural logarithm integration straightforward, embodying the power of strategic substitution in calculus.

Natural Logarithm Integration

Integration involving natural logarithms arises commonly when dealing with functions of the form \(\frac{1}{x}\). The integral of \(\frac{1}{x}\) is \(\ln|x|\), a staple in calculus for dealing with continuous growth or decay.
For both Alicia's and Boris’s velocity functions, after applying u-substitution, it boiled down to integrating \(\int \frac{1}{u} du\). This directly results in a natural logarithm:

• \(\int \frac{1}{u} du = \ln|u| + C\)

Thus, the emergence of logarithmic forms in their position formulas, like \(s_A(t) = 4\ln|(t+1)|\), showcases this fundamental principle.
Natural logarithm integration is vital for understanding phenomena exhibiting exponential decay or growth, commonly encountered across various scientific fields.