/*! 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 70 Find the solution of the followi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 70

Find the solution of the following initial value problems. $$h^{\prime}(t)=6 \sin 3 t ; h(\pi / 6)=6$$

Short Answer

Expert verified
Question: Find the function h(t) given that h'(t) = 6sin(3t) and h(蟺/6) = 6. Answer: The function h(t) is h(t) = 2(-cos(3t)) + 6.

Step by step solution

01

Integrate h'(t) with respect to t

We are given h'(t) = 6sin(3t). We need to integrate this function with respect to t to find h(t): $$h(t) = \int 6 \sin(3t) dt$$ We will use a simple substitution for this integration where u = 3t: $$du = 3 dt$$ $$\frac{1}{3}du = dt$$ Now we can rewrite the integral: $$h(t) = \int 6 \sin(u) \frac{1}{3} du$$ $$h(t) = 2 \int \sin(u) du$$ Now, we can integrate sin(u) with respect to u: $$h(t) = 2(-\cos(u)) + C$$ Substitute back u = 3t: $$h(t) = 2(-\cos(3t)) + C$$
02

Use the initial condition to find the constant of integration

We are given that h(蟺/6) = 6. Plug in the values of t and h(t) to find the constant C: $$6 = 2(-\cos(3(\pi/6))) + C$$ $$6 = 2(-\cos(\pi/2)) + C$$ Since cos(蟺/2) = 0, $$6 = 2(0) + C$$ $$C = 6$$
03

Write the final solution for h(t)

Now we can write the final solution for h(t) by substituting the constant of integration C: $$h(t) = 2(-\cos(3t)) + 6$$

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

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32 ; v(0)=20, s(0)=0$$

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation \(a(t)=v^{\prime}(t)=g,\) where \(g=-9.8 \mathrm{m} / \mathrm{s}^{2}\). a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A stone is thrown vertically upward with a velocity of \(30 \mathrm{m} / \mathrm{s}\) from the edge of a cliff 200 m above a river.

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=\sin ^{-1} x$$

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(\log _{2} x-\log _{3} x\right)$$

Show that any exponential function \(b^{x},\) for \(b>1,\) grows faster than \(x^{p},\) for \(p>0\).
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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.