/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 Write and solve the differential... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 9

Write and solve the differential equation that models the verbal statement. The rate of change of \(Q\) with respect to \(t\) is inversely proportional to the square of \(t\).

Short Answer

Expert verified
The differential equation that models the given verbal statement is: dQ/dt = k/t虏. The solution to this differential equation is: Q = -k/t + C, where k and C are constants.

Step by step solution

01

Write the Differential Equation

From the statement, 'The rate of change of Q with respect to t is inversely proportional to the square of t', we can derive the mathematical equivalent. The rate change of Q with respect to t can be represented as dQ/dt. 'Inversely proportional to the square of t' can be represented as 1/t虏. Therefore, the differential equation that models the verbal statement is given as: dQ/dt = k/t虏, where k is a constant.
02

Solve the Differential Equation

This is a separable differential equation which can be rearranged and solved by integrating both sides. The equation can be rewritten as dQ = k (1/t虏) dt. Then, integrate both sides of the equation. For the left side, the integral of dQ is just Q. For the right side, integrating k (1/t虏) dt gives -k/t since the integral of t^-2 is -t^-1. Therefore, upon integration, the solution to the differential equation is: Q = -k/t + C, where C is the constant of integration.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ y=x^{3}-2 x, \quad(-1,1) $$

The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\). Explain why this is so. Use symmetry to write a single integral that does represent the area.

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\sqrt{2 x}, \quad y=x^{2} $$

The chief financial officer of a company reports that profits for the past fiscal year were \(\$ 893,000\). The officer predicts that profits for the next 5 years will grow at a continuous annual rate somewhere between \(3 \frac{1}{2} \%\) and \(5 \%\). Estimate the cumulative difference in total profit over the 5 years based on the predicted range of growth rates.

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=\arctan x, \quad y=0, \quad x=0, \quad x=1\) (a) 10 (b) \(\frac{3}{4}\) (c) 5 (d) -6 (e) 15
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.