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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.74a
Chapter 7, Problem 7.3.74a

Wave velocity Use Exercise 73 to do the following calculations.
a. Find the velocity of a wave where λ = 50 m and d = 20 m.

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Identify the formula for wave velocity, which is given by \(v = f \times \lambda\), where \(v\) is the velocity, \(f\) is the frequency, and \(\lambda\) is the wavelength.
Note that the problem provides the wavelength \(\lambda = 50\) m, but the frequency \(f\) is not directly given. Instead, you have a distance \(d = 20\) m, which likely relates to the wave's period or time information from Exercise 73.
Recall that frequency \(f\) is the reciprocal of the period \(T\), i.e., \(f = \frac{1}{T}\). If Exercise 73 provides the time it takes for the wave to travel distance \(d\), you can find the period or frequency from that data.
Calculate the frequency \(f\) using the information from Exercise 73 (for example, if the wave takes time \(t\) to travel distance \(d\), then \(f = \frac{v}{\lambda}\) or use \(f = \frac{1}{T}\) if period \(T\) is known).
Finally, substitute the values of \(f\) and \(\lambda\) into the wave velocity formula \(v = f \times \lambda\) to find the velocity of the wave.

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Key Concepts

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

Wave Velocity Formula

Wave velocity is the speed at which a wave propagates through a medium. It is commonly calculated using the formula v = f × λ, where v is velocity, f is frequency, and λ (lambda) is the wavelength. Understanding this relationship is essential to find wave velocity when wavelength and frequency are known.
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Wavelength (λ)

Wavelength is the distance between two consecutive points in phase on a wave, such as crest to crest or trough to trough. It is usually measured in meters and is a key parameter in determining wave velocity and frequency. Knowing the wavelength helps relate spatial properties of the wave to its speed.

Frequency and its Relation to Distance (d)

Frequency is the number of wave cycles passing a point per second, measured in hertz (Hz). In some problems, distance (d) may relate to the wave’s period or frequency, such as the distance traveled in one period. Understanding how distance and time relate helps find frequency when it is not directly given.
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Related Practice
Textbook Question

Visual approximation


a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.

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Textbook Question

Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

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Textbook Question

Shallow-water velocity equation

a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.

Textbook Question

A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.


a. Show that a satisfies the equation cosh 50/a − 1 = 10/a.

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Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.

Textbook Question

Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.


a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.