Domanda del libro di testo
Visual approximation
a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.
42
views
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testo
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.95a
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.95aCapitolo 7, Problema 7.3.95a
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.
Guida verificata passo dopo passo1
Identify the given function for the distance fallen: \(d(t) = \frac{m}{k} \ln \left( \cosh \left( \sqrt{\frac{kg}{m}} \, t \right) \right)\), where \(m\) is mass, \(k\) is a constant, \(g = 9.8\) m/s², and \(t\) is time in seconds.
Substitute the known values into the formula: \(m = 75\), \(k = 0.2\), \(g = 9.8\), and \(t = 10\) seconds.
Calculate the term inside the square root: \(\sqrt{\frac{kg}{m}} = \sqrt{\frac{0.2 \times 9.8}{75}}\).
Evaluate the argument of the hyperbolic cosine function: \(\sqrt{\frac{kg}{m}} \times t = \left( \text{value from previous step} \right) \times 10\).
Compute the distance fallen by plugging the value into the formula: \(d(10) = \frac{75}{0.2} \times \ln \left( \cosh \left( \text{value from previous step} \right) \right)\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Modeling Air Resistance in Free Fall
Air resistance affects falling objects by opposing motion, often modeled as proportional to velocity squared for high speeds. This nonlinear drag force changes the acceleration and velocity over time, making the motion differ from simple free fall. Understanding this helps interpret the given distance formula involving hyperbolic functions.
Video consigliato:
09:29
Exponential Growth & Decay
Hyperbolic Functions and Their Properties
Hyperbolic functions like cosh(x) and sinh(x) arise in solutions to differential equations involving quadratic velocity terms. The function cosh(x) = (e^x + e^{-x})/2 grows exponentially and appears in the distance formula, reflecting the balance between gravity and air resistance in the falling motion.
Video consigliato:
Percorso guidato
06:21Properties of Functions
Applying Given Formulas with Physical Constants
To find the distance fallen, substitute the known values (mass m, gravity g, constant k, and time t) into the formula d(t) = (m/k) ln(cosh(√(kg/m) t)). This requires careful calculation of the square root term and the natural logarithm, ensuring units are consistent and the physical meaning is preserved.
Video consigliato:
08:29Example 5: Packaging Design
Pratica correlata
Domanda del libro di testo
Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.
a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.
47
views
Domanda del libro di testo
Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4
64
views
Domanda del libro di testo
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.
45
views
Domanda del libro di testo
Terminal velocity Refer to Exercises 95 and 96.
a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).
65
views
Domanda del libro di testo
Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
a. cosh 0
59
views