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To convert the rectangular coordinates ( A(2, 3, 1) ) into cylindrical coordinates, we use the following formulas:
1. ( r = sqrt{x^2 + y^2} )
2. ( theta = tan^{-1}(frac{y}{x}) )
3. ( z = z )
Where ( (x, y, z) ) are the rectangular coordinates.
Given ( A(2, 3, 1) ):
1. Calculate ( r ):
[
r = sqrt{2^2 + 3^2} = sqrt{4 + 9} = sqrt{13}
]
2. Calculate ( theta ):
[
theta = tan^{-1}(frac{3}{2}) approx 0.9828 text{ radians} quad text{(approximately 56.31 degrees)}
]
3. The ( z ) coordinate remains the same:
[
z = 1
]
Thus, the cylindrical coordinates are ( (r, theta, z) ):
[
left( sqrt{13}, tan^{-1}left(frac{3}{2}right), 1 right)
]
Therefore, the cylindrical coordinates corresponding to the point ( A(2, 3, 1) ) are approximately:
Answer: d
Explanation: ρ = √(x2+y2) = √13 = 3.61
Φ = tan-1(y/x) = 56.31
z = 1
Thus, A = (3.61,56.31,1).