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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.6.57b
Chapter 3, Problem 3.6.57b

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>
b. Which runner has the greater average angular velocity?

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To find the average angular velocity of each runner, we need to use the formula for average angular velocity, which is the total change in angular position divided by the total time taken. Mathematically, this is expressed as: \( \text{Average Angular Velocity} = \frac{\Delta \theta}{\Delta t} \) for Jean and \( \frac{\Delta \phi}{\Delta t} \) for Juan.
Determine the total change in angular position for each runner. Since both start at \( \theta = 0 \) and \( \phi = 0 \) and finish at \( \theta = 2\pi \) and \( \phi = 2\pi \), the change in angular position for both runners is \( 2\pi \) radians.
The total time for the race is given as \( \Delta t = 4 \) minutes.
Calculate the average angular velocity for Jean using the formula: \( \text{Average Angular Velocity for Jean} = \frac{2\pi}{4} \).
Calculate the average angular velocity for Juan using the formula: \( \text{Average Angular Velocity for Juan} = \frac{2\pi}{4} \). Compare the two values to determine which runner has the greater average angular velocity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Position

Angular position refers to the angle that an object has rotated from a reference point, typically measured in radians. In the context of the race, θ(t) and ϕ(t) represent the angular positions of Jean and Juan over time, indicating their respective locations on the circular track. Understanding angular position is crucial for determining how far each runner has traveled around the track.
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Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a point, expressed as the rate of change of angular position with respect to time. It is denoted as θ′(t) for Jean and ϕ′(t) for Juan. To compare the average angular velocities of the two runners, one must calculate the change in their angular positions over the time interval, which provides insight into their speed around the track.
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Average Angular Velocity

Average angular velocity is defined as the total change in angular position divided by the total time taken. For the runners, it can be calculated using the formula (θ(final) - θ(initial)) / (t(final) - t(initial)) for Jean and similarly for Juan. This concept is essential for determining which runner has a greater average angular velocity over the course of the race, allowing for a direct comparison of their performances.
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Related Practice
Textbook Question

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

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b. Find the slope of the curve at the given point.

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b. Determine an equation of the line tangent to the curve at the given point.

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b. How fast will the city be growing when it reaches a size of 38 mi²?

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