Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 25

Capitolo 4, Problema 25

As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)

Guida verificata passo dopo passo

First, convert the total increase in volume from acre-feet to gallons. Since 1 acre-foot is 43,560 cubic feet and 1 cubic foot holds 7.48 gallons, multiply 1400 acre-feet by 43,560 to get the volume in cubic feet, and then multiply by 7.48 to convert to gallons.

Next, calculate the average rate of increase in gallons per minute over the 24-hour period. There are 24 hours in a day and 60 minutes in an hour, so multiply 24 by 60 to find the total number of minutes in the period.

Divide the total increase in gallons by the total number of minutes to find the average rate of increase in gallons per minute.

Apply the Mean Value Theorem for derivatives, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval.

Since the average rate of increase is less than 225,000 gal/min, and the total increase is 1400 acre-ft, use the Mean Value Theorem to conclude that there must be at least one instant during the 24-hour period where the rate of increase exceeded 225,000 gal/min.

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:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Rate of Change

In calculus, the rate of change refers to how a quantity changes over time. It is often represented as a derivative, which measures the instantaneous rate of change of a function. In this context, we are interested in how the volume of water in the reservoir changes with respect to time, specifically looking for the maximum rate of increase during the 24-hour period.

Video consigliato:

Percorso guidato

04:16

Intro To Related Rates

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, there exists at least one point where the instantaneous rate of change (derivative) equals the average rate of change over that interval. This theorem is crucial for this problem as it allows us to conclude that there must be a moment when the volume increase exceeds the average rate calculated over the 24 hours.

Video consigliato:

Percorso guidato

06:11

Fundamental Theorem of Calculus Part 1

Unit Conversion

Unit conversion is the process of converting a quantity expressed in one set of units to another. In this problem, we need to convert the increase in volume from acre-feet to gallons per minute to compare it with the required rate. Understanding how to convert between these units is essential for accurately determining whether the instantaneous rate exceeds 225,000 gallons per minute.

Video consigliato:

5:10

Evaluate Composite Functions - Values on Unit Circle

Pratica correlata

Domanda del libro di testo

Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?

208

views

Domanda del libro di testo

Finding Functions from Derivatives

Suppose that f'(x) = 2x for all x. Find f(2) if

a. f(0) = 0

190

views

Domanda del libro di testo

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y = (x² - 49) / (x² + 5x - 14)

111

views

Domanda del libro di testo

y=1-(x+1)^3

198

views

Domanda del libro di testo

The Mean Value Theorem

a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

[Technology Exercises] b.Find the solution to as many decimal places as you can.

277