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

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, \[ (7.3,\,3.9,\,1.3) \]