/*! 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 63 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 63

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r^{2}=16 \sin 2 \theta$$

Short Answer

Expert verified
The graph is traced only once for \(\theta\) in the interval \([0, 2\pi]\).

Step by step solution

01

Rewrite the equation in x and y

The polar equation \(r^{2}=16 \sin 2 \theta\) can be rewritten using the Cartesian form \(x = r \cos \(\theta\) \) and \(y = r \sin \(\theta\)\) as \(x^2 + y^2 = 16y\).
02

Draw the Cartesian graph

Use a graphing utility to plot the equation \(x^2 + y^2 = 16y\) in a graph. This equation represents a circle with its center at (0, 8) and radius 8.
03

Find the interval for theta

Now, notice that the circle starts at the point (0,16) when \(\theta = 0\) and traces the circle until it reaches the starting point again when \(\theta\) equals one complete rotation, i.e., \(2\pi\). Thus, the interval for \(\theta\) where the graph is traced only once is \([0, 2\pi]\).

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

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=2 \sin \theta$$

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((\mathrm{c})\) find the area of the triangle. $$A(-3,0), B(0,-2), C(2,3)$$

A quarterback releases a pass at a height of 7 feet above the playing field, and a receiver catches the football at a height of 4 feet,30 yards directly downfield. The pass is released at an angle of \(35^{\circ}\) with the horizontal. (a) Write a set of parametric equations for the path of the football. (See Exercises 93 and 94 .) (b) Find the speed of the football when it is released. (c) Use a graphing utility to graph the path of the football and approximate its maximum height. (d) Find the time the receiver has to position himself after the quarterback releases the football.

Consider a line with slope \(m\) and \(y\) -intercept \((0,4)\) (a) Write the distance \(d\) between the point \((3,1)\) and the line as a function of \(m\) (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the point and the line. (d) Is it possible for the distance to be \(0 ?\) If so, what is the slope of the line that yields a distance of \(0 ?\) (e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

Find the distance between the point and the line. Point \((-2,6)\) Line \(y=-x+5\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.