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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 105b
Capitolo 3, Problema 105b

Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.


b. How many revolutions per minute is 0.6 rad/sec?


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Guida verificata passo dopo passo
1
Understand the problem: We need to convert the angular speed from radians per second to revolutions per minute.
Recall the relationship between radians and revolutions: One complete revolution is equivalent to 2π radians.
Calculate the number of revolutions per second: Divide the angular speed by 2π to convert from radians per second to revolutions per second. Use the formula: \( \text{revolutions per second} = \frac{d\theta/dt}{2\pi} \).
Convert revolutions per second to revolutions per minute: Multiply the result from the previous step by 60, since there are 60 seconds in a minute.
Express the final result: The number of revolutions per minute is the product of the conversion factor and the angular speed in revolutions per second.

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Radians and Revolutions

Radians are a unit of angular measure used in mathematics, where one full revolution (360 degrees) is equal to 2π radians. Understanding the relationship between radians and revolutions is crucial for converting angular velocity from radians per second to revolutions per minute.
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Converting between Degrees & Radians

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or moves around a central point, typically expressed in radians per second. In this context, the angular velocity of the searchlight beam is given as -0.6 rad/sec, indicating the rate at which the beam sweeps across the shore.
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Derivatives Applied To Velocity

Conversion of Units

Converting units is a fundamental skill in calculus and physics, allowing for the translation of measurements from one system to another. To find revolutions per minute from radians per second, one must apply conversion factors, specifically recognizing that 1 revolution equals 2π radians and there are 60 seconds in a minute.
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Evaluate Composite Functions - Values on Unit Circle
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Resistors connected in parallel If two resistors of R₁ and R₂ ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation


1/R = 1/R₁ + 1/R₂


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If R₁ is decreasing at the rate of 1ohm/sec and R₂ is increasing at the rate of 0.5 ohm/sec, at what rate is R changing when R₁ = 75 ohms and R₂ = 50 ohms?

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Find the linearizations of


a. tan x at x = -π/4


Graph the curves and linearizations together.

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Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation

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S = πr √ r² + h².


c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

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How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

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Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

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Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.


a. What is the relation between the variables h and r in the figure?


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