/*! 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 19 Two points \(A\) and \(B\) move ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 19

Two points \(A\) and \(B\) move along the \(x\) -axis. After \(t\) sec, their positions are given by the equations $$ \begin{array}{ll} A: & x=3 t+100 \\ B: & x=20 t-36 \end{array} $$ (a) Which point is traveling faster, \(A\) or \(B ?\) (b) Which point is farther to the right when \(t=0 ?\) (c) At what time \(t\) do \(A\) and \(B\) have the same \(x\) -coordinate?

Short Answer

Expert verified
(a) Point B; (b) Point A; (c) t = 8 seconds.

Step by step solution

01

Determine Speeds

The speed of a point moving along the x-axis is given by the coefficient of \(t\) in the equation of motion. For point \(A\), \(x_A = 3t + 100\), the speed is 3 units/sec. For point \(B\), \(x_B = 20t - 36\), the speed is 20 units/sec. Thus, point \(B\) is traveling faster.
02

Calculate Initial Positions

To find which point is farther to the right at \(t = 0\), substitute \(t = 0\) in both equations. For point \(A\), \(x_A = 3(0) + 100 = 100\). For point \(B\), \(x_B = 20(0) - 36 = -36\). Since 100 is greater than -36, point \(A\) is farther to the right at \(t = 0\).
03

Set Equations Equal to Find Equal Coordinate

Set the equations for \(A\) and \(B\) equal to each other to find when they have the same x-coordinate: \(3t + 100 = 20t - 36\). Simplify to \(100 + 36 = 20t - 3t\), which gives \(136 = 17t\). Solve for \(t\): \(t = \frac{136}{17} = 8\) seconds.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed Calculation
When discussing the motion of objects along a straight path, such as the x-axis, speed is a key factor. Speed defines how fast an object is moving and is expressed in units of distance per unit of time, such as meters per second or units per second.
To calculate the speed of a point moving along the x-axis, we can look at the coefficient of the time variable, \( t \), in its position equation. In the given exercise, point \( A \) has the position equation \( x_A = 3t + 100 \). The coefficient of \( t \) here is 3, indicating point \( A \) moves at a speed of 3 units per second.
Similarly, point \( B \) has the position equation \( x_B = 20t - 36 \). The coefficient of \( t \) in this equation is 20, revealing that \( B \) travels at a speed of 20 units per second.
  • Thus, point \( B \) travels faster than point \( A \).
Coordinate Equations
Coordinate equations are mathematical expressions that help us determine the position of a moving object along a particular path at any given time. These equations are often given in the form of a linear equation when describing one-dimensional motion, such as along the x-axis.
The coordinate equations for points \( A \) and \( B \) in our exercise are:
  • For point \( A \): \( x = 3t + 100 \)
  • For point \( B \): \( x = 20t - 36 \)
To find the position of these points when time \( t = 0 \), we simply substitute 0 in place of \( t \) in each equation:
  • For \( A \), when \( t = 0 \), \( x_A = 100 \)
  • For \( B \), when \( t = 0 \), \( x_B = -36 \)
From this calculation, we see that at time zero, \( A \) is positioned at 100 units along the x-axis, while \( B \) is at -36 units, putting \( A \) farther to the right.
Simultaneous Coordinates
In situations where two moving objects might cross paths or align with each other, it is useful to determine when they will have the same coordinates. This involves solving the coordinate equations simultaneously for the same value of \( x \).
For points \( A \) and \( B \), we want to find when their x-coordinates are equal. This requires setting their position equations equal to each other: \[ 3t + 100 = 20t - 36 \].
Solving the equation involves:
  • Bringing like terms to one side: \( 100 + 36 = 20t - 3t \)
  • Simplifying gives: \( 136 = 17t \)
  • Solving for \( t \), we get: \( t = \frac{136}{17} = 8 \)
Thus, points \( A \) and \( B \) will have the same x-coordinate at \( t = 8 \) seconds. At this moment, the two points are exactly aligned along the x-axis.

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

(a) Determine the \(x\) - and \(y\) -intercepts and the excluded regions for the graph of the given function. Specify your results using a sketch similar to Figure \(16(a) .\) In Exercises \(31-34\) you will first need to factor the polynomial. (b) Graph each function. $$y=x^{3}+3 x^{2}-4 x-12$$

This exercise shows that if we have a table generated by a linear function and the \(x\) -values are equally spaced, then the first differences of the \(y\) -values are constant. (a) In the following data table, the three \(x\) -entries are equally spaced. Compute the three entries in the \(f(x)\) row assuming that \(f\) is the linear function given by \(f(x)=m x+b\) \(f(x)=m x+b .\) (Don't worry about the fact that your answers contain all four of the letters \(m, b, a, \text { and } h .)\) \begin{tabular}{llll} \hline\(x\) & \(a\) & \(a+h\) & \(a+2 h\) \\ \(f(x)\) & & & \\ \hline \end{tabular} (b) Compute the first differences for the three quantities that you listed in the \(f(x)\) row in part (a). (The two first differences that you obtain should turn out to be equal, as required.)

Let \(f(x)=x^{2} .\) Find the average rate of change \(\Delta f / \Delta x\) on the interval \([a, x]\).

The point \(P\) is in the first quadrant on the graph of \(y=1-x^{4} .\) From \(P,\) perpendiculars are drawn to the \(x\) - and \(y\) -axes, thus forming a rectangle. (a) Express the area of the rectangle as a function of a single variable. (b) Use a graphing utility to graph the area function. Then, using the zoom feature, estimate to the nearest hundredth the maximum possible area for the rectangle.

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$f \circ g$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.