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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 7.3.96c
Chapter 7, Problem 7.3.96c

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.


c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

Verified step by step guidance
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Recall that the velocity function \(v(t)\) is the derivative of the position function \(s(t)\) with respect to time \(t\). So, start by writing down the velocity function \(v(t) = s'(t)\).
From Exercise 95, the position function for a falling body with mass \(m\) and drag coefficient \(k\) is typically given by \(s(t) = \frac{mg}{k}t - \frac{m^2g}{k^2}(1 - e^{-\frac{k}{m}t})\). Differentiate this function to find the velocity function \(v(t)\).
After differentiating, you should get \(v(t) = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\). This formula shows how velocity changes over time considering air resistance.
Set the velocity equal to 45 m/s to find the time \(t\) when the BASE jumper reaches this speed: \(45 = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\).
Solve the equation for \(t\) by isolating the exponential term, then taking the natural logarithm: first, rearrange to get \(e^{-\frac{k}{m}t} = 1 - \frac{45k}{mg}\), then take \(\ln\) on both sides and solve for \(t\).

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Key Concepts

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

Position and Velocity Functions

The position function describes the location of a falling body over time, while the velocity function is its derivative, representing the rate of change of position. Understanding how to differentiate the position function is essential to find the velocity at any given time.
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Using The Velocity Function

Solving for Time Given Velocity

To find the time when the velocity reaches a specific value, set the velocity function equal to that value and solve for time. This often involves algebraic manipulation or solving transcendental equations depending on the form of the velocity function.
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Parameters in Motion Equations (Mass and Drag Coefficient)

Mass (m) and drag coefficient (k) affect the motion of the falling body by influencing acceleration and terminal velocity. These parameters appear in the differential equations governing motion and must be used correctly to model realistic velocity and position.
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Related Practice
Textbook Question

Acceleration, velocity, position Suppose the acceleration of an object moving along a line is given by a(t) = -k v(t), where k is a positive constant and v is the object's velocity. Assume the initial velocity and position are given by v(0) = 10 and s(0) = 0, respectively.

c. Use the fact that dv/dt = (dv/ds)(ds/dt) (by the Chain Rule) to find the velocity as a function of position.

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Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

Textbook Question

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.

c. csch⁻¹ 5

Textbook Question

Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as

x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.



b. In Chapter 8, the formula for the integral in part (a) is derived:

∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.

Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:

t = ln(x + √(x² − 1)).

Textbook Question

Many formulas There are several ways to express the indefinite integral of sech x.


b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Textbook Question

Power lines 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.

c. Use your answer in part (b) to find a, and then compute the length of the power line.