/*! 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 Express the following polar coor... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 19

Express the following polar coordinates in Cartesian coordinates. $$\left(-4, \frac{3 \pi}{4}\right)$$

Short Answer

Expert verified
Question: Convert the polar coordinates \(\left(-4, \frac{3 \pi}{4}\right)\) to Cartesian coordinates. Answer: \(\left( 2\sqrt{2}, -2\sqrt{2} \right)\)

Step by step solution

01

Identify the given polar coordinates

We are given $$\left(-4, \frac{3 \pi}{4}\right),$$ where \(r = -4\) and \(\theta = \frac{3 \pi}{4}\).
02

Convert the polar coordinates to Cartesian coordinates

Use the equations for converting polar to Cartesian coordinates: $$ x = r \cos \theta = -4 \cos \frac{3 \pi}{4} $$ $$ y = r \sin \theta = -4 \sin \frac{3 \pi}{4} $$
03

Calculate the Cartesian coordinates

Find the cosine and sine values and multiply by \(r\): $$ x = -4 \left( -\frac{1}{\sqrt{2}} \right) = \frac{4}{\sqrt{2}} $$ $$ y = -4 \left( \frac{1}{\sqrt{2}} \right) = -\frac{4}{\sqrt{2}} $$
04

Simplify the Cartesian coordinates

To simplify the Cartesian coordinates, multiply the numerator and denominator by \(\sqrt{2}\) to rationalize the denominator: $$ x = \frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = 2\sqrt{2} $$ $$ y = -\frac{4}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = -2\sqrt{2} $$
05

Write the final answer in Cartesian coordinates

The Cartesian coordinates are: $$ \left( 2\sqrt{2}, -2\sqrt{2} \right) $$

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Most popular questions from this chapter

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The hyperbola \(x^{2} / 4-y^{2} / 9=1\) has no \(y\) -intercepts. b. On every ellipse, there are exactly two points at which the curve has slope \(s,\) where \(s\) is any real number. c. Given the directrices and foci of a standard hyperbola, it is possible to find its vertices, eccentricity, and asymptotes. d. The point on a parabola closest to the focus is the vertex.

Water flows in a shallow semicircular channel with inner and outer radii of \(1 \mathrm{m}\) and \(2 \mathrm{m}\) (see figure). At a point \(P(r, \theta)\) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on \(r\), the distance from the center of the semicircles. a. Express the region formed by the channel as a set in polar coordinates. b. Express the inflow and outflow regions of the channel as sets in polar coordinates. c. Suppose the tangential velocity of the water in \(\mathrm{m} / \mathrm{s}\) is given by \(v(r)=10 r,\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.5, \frac{\pi}{4}\right)\) or \(\left(1.2, \frac{3 \pi}{4}\right) ?\) Explain. d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r},\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.8, \frac{\pi}{6}\right)\) or \(\left(1.3, \frac{2 \pi}{3}\right) ?\) Explain. e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_{0}\) ) is proportional to \(\int_{1}^{2} v(r) d r .\) Is the total flow through the channel greater for the flow in part (c) or (d)?

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1+2 \cos \theta}$$

Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$10 x^{2}-7 y^{2}=140$$

Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)
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