Question
upstudy study bank question image url

enovo McAfee YouTube Gmail Maps Cart - Cracker Barrel Question 9 \( 0 / 1 \) pt \( 3 \stackrel{\leftrightarrows}{\rightleftarrows} 99 \) Details Convert the point \( (x, y, z)=(-5,-5,2) \) to spherical coordinates. Give answers as positive values, either as expressions, or decimals to one decimal place. \[ (\rho, \theta, \phi)= \] \( \square \) Question Help: Video Submit Question

Ask by Davies Gardner. in the United States
Mar 08,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ (\rho, \theta, \phi) = (7.3, 3.9, 1.3) \]

Solution

We start with the Cartesian point \[ (x,y,z)=(-5,-5,2) \] In spherical coordinates, the point is given by \[ (\rho,\theta,\phi) \] where 1. \(\rho=\sqrt{x^2+y^2+z^2}\) 2. \(\theta=\) the angle in the \(xy\)-plane (measured from the positive \(x\)-axis) 3. \(\phi=\) the angle from the positive \(z\)-axis --- **Step 1. Find \(\rho\):** \[ \rho=\sqrt{(-5)^2+(-5)^2+2^2}=\sqrt{25+25+4}=\sqrt{54}=3\sqrt{6} \approx 7.3 \] --- **Step 2. Find \(\theta\):** Using \[ \theta=\arctan\left(\frac{y}{x}\right) \] we get \[ \arctan\left(\frac{-5}{-5}\right)=\arctan(1)=\frac{\pi}{4} \] However, note that the point \((-5,-5)\) lies in the third quadrant where both \(x\) and \(y\) are negative. To account for the correct quadrant, we add \(\pi\): \[ \theta=\frac{\pi}{4}+\pi=\frac{5\pi}{4} \approx 3.9 \] --- **Step 3. Find \(\phi\):** \[ \phi=\arccos\left(\frac{z}{\rho}\right)=\arccos\left(\frac{2}{3\sqrt{6}}\right) \] To approximate, first compute: \[ 3\sqrt{6}\approx 7.3,\quad \frac{2}{7.3}\approx 0.274 \] Then, \[ \phi\approx\arccos(0.274)\approx 1.3 \] --- **Answer:** \[ (\rho,\theta,\phi)=\left(3\sqrt{6}, \frac{5\pi}{4}, \arccos\left(\frac{2}{3\sqrt{6}}\right)\right) \] or, approximated in decimals, \[ (7.3,\,3.9,\,1.3) \]

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Extra Insights

To convert the Cartesian coordinates \((-5, -5, 2)\) to spherical coordinates \((\rho, \theta, \phi)\), we first calculate \(\rho\), which is the distance from the origin. It's given by the formula \(\rho = \sqrt{x^2 + y^2 + z^2}\). Plugging in the values: \(\rho = \sqrt{(-5)^2 + (-5)^2 + 2^2} = \sqrt{25 + 25 + 4} = \sqrt{54} \approx 7.35\). Next, we determine \(\theta\), which is the angle in the \(xy\)-plane from the positive \(x\)-axis. It is calculated using \(\tan(\theta) = \frac{y}{x}\), so \(\theta = \tan^{-1}\left(\frac{-5}{-5}\right) = \tan^{-1}(1) = \frac{3\pi}{4}\) (adjusted to the correct quadrant, where both \(x\) and \(y\) are negative). For \(\phi\), the polar angle from the positive \(z\)-axis, we use the formula \(\phi = \cos^{-1}\left(\frac{z}{\rho}\right)\): \(\phi = \cos^{-1}\left(\frac{2}{\sqrt{54}}\right) \approx 1.36\) radians. Thus, the spherical coordinates are: \[ \left(\sqrt{54}, \frac{5\pi}{4}, 1.36\right) \]

Related Questions

Latest Pre Calculus Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy