Questions: Express the point given in Cartesian coordinates in cylindrical coordinates (r, θ, z). A) (9(1/2), 9(sqrt(3)/2),-6)=() () () B) (-9(1/2), 9(sqrt(3)/2),-6)=() () () C) (9(1/2),-9(sqrt(3)/2),-6)=() () () D) (-9(1/2),-9(sqrt(3)/2),-6)=() () ()

Express the point given in Cartesian coordinates in cylindrical coordinates (r, θ, z).
A) (9(1/2), 9(sqrt(3)/2),-6)=()
()
()
B) (-9(1/2), 9(sqrt(3)/2),-6)=()
()
()
C) (9(1/2),-9(sqrt(3)/2),-6)=()
()
()
D) (-9(1/2),-9(sqrt(3)/2),-6)=()
()
()
Transcript text: (1 point) Express the point given in Cartesian coordinates in cylindrical coordinates $(r, \theta, z)$. A) $\left(9\left(\frac{1}{2}\right), 9\left(\frac{\sqrt{3}}{2}\right),-6\right)=($ $\square$ $\square$ $\square$ ) B) $\left(-9\left(\frac{1}{2}\right), 9\left(\frac{\sqrt{3}}{2}\right),-6\right)=($ $\square$ $\square$ $\square$ ) C) $\left(9\left(\frac{1}{2}\right),-9\left(\frac{\sqrt{3}}{2}\right),-6\right)=($ $\square$ $\square$ $\square$ ) D) $\left(-9\left(\frac{1}{2}\right),-9\left(\frac{\sqrt{3}}{2}\right),-6\right)=($ $\square$ $\square$ $\square$ )
failed

Solution

failed
failed

Solution Steps

To convert Cartesian coordinates (x,y,z)(x, y, z) to cylindrical coordinates (r,θ,z)(r, \theta, z), we use the following formulas:

  1. r=x2+y2 r = \sqrt{x^2 + y^2}
  2. θ=atan2(y,x) \theta = \text{atan2}(y, x)
  3. The z z -coordinate remains the same.

We will apply these formulas to each of the given points.

Step 1: Convert Point A

For point A, given the Cartesian coordinates (9(12),9(32),6) \left(9\left(\frac{1}{2}\right), 9\left(\frac{\sqrt{3}}{2}\right), -6\right) :

  • Calculate r r : r=(9(12))2+(9(32))2=9.0 r = \sqrt{\left(9\left(\frac{1}{2}\right)\right)^2 + \left(9\left(\frac{\sqrt{3}}{2}\right)\right)^2} = 9.0
  • Calculate θ \theta : θ=atan2(9(32),9(12))=1.0472 radians \theta = \text{atan2}\left(9\left(\frac{\sqrt{3}}{2}\right), 9\left(\frac{1}{2}\right)\right) = 1.0472 \text{ radians}
  • The z z -coordinate remains the same: z=6 z = -6 .

Thus, the cylindrical coordinates for point A are (9.0,1.0472,6) (9.0, 1.0472, -6) .

Step 2: Convert Point B

For point B, given the Cartesian coordinates (9(12),9(32),6) \left(-9\left(\frac{1}{2}\right), 9\left(\frac{\sqrt{3}}{2}\right), -6\right) :

  • Calculate r r : r=(9(12))2+(9(32))2=9.0 r = \sqrt{\left(-9\left(\frac{1}{2}\right)\right)^2 + \left(9\left(\frac{\sqrt{3}}{2}\right)\right)^2} = 9.0
  • Calculate θ \theta : θ=atan2(9(32),9(12))=2.0944 radians \theta = \text{atan2}\left(9\left(\frac{\sqrt{3}}{2}\right), -9\left(\frac{1}{2}\right)\right) = 2.0944 \text{ radians}
  • The z z -coordinate remains the same: z=6 z = -6 .

Thus, the cylindrical coordinates for point B are (9.0,2.0944,6) (9.0, 2.0944, -6) .

Step 3: Convert Point C

For point C, given the Cartesian coordinates (9(12),9(32),6) \left(9\left(\frac{1}{2}\right), -9\left(\frac{\sqrt{3}}{2}\right), -6\right) :

  • Calculate r r : r=(9(12))2+(9(32))2=9.0 r = \sqrt{\left(9\left(\frac{1}{2}\right)\right)^2 + \left(-9\left(\frac{\sqrt{3}}{2}\right)\right)^2} = 9.0
  • Calculate θ \theta : θ=atan2(9(32),9(12))=1.0472 radians \theta = \text{atan2}\left(-9\left(\frac{\sqrt{3}}{2}\right), 9\left(\frac{1}{2}\right)\right) = -1.0472 \text{ radians}
  • The z z -coordinate remains the same: z=6 z = -6 .

Thus, the cylindrical coordinates for point C are (9.0,1.0472,6) (9.0, -1.0472, -6) .

Final Answer

  • Point A: (9.0,1.0472,6) \boxed{(9.0, 1.0472, -6)}
  • Point B: (9.0,2.0944,6) \boxed{(9.0, 2.0944, -6)}
  • Point C: (9.0,1.0472,6) \boxed{(9.0, -1.0472, -6)}
Was this solution helpful?
failed
Unhelpful
failed
Helpful