The cylindrical coordinates of a point are (2√3, π/3, 2). find the rectangular and spherical coordinates of the point.

(x ,y ,z) = (√3 ,3 ,2) (rectangular coordinates)

(ρ, θ, φ) = (4 , π/3, π/3) (spherical coordinates)

Step-by-step explanation:

if the cylindrical coordinates are

( r , θ , z) = (2√3, π/3, 2)

then assuming that the θ angle is the one in the x-y plane relative to the x axis :

x= r*cos  θ = 2√3*cos (π/3) = 2√3* (1/2)= √3

y= r*sin  θ = 2√3*sin (π/3) = 2√3* (√3/2) = 3

z = z =2

then (x ,y ,z) = (√3 ,3 ,2) (rectangular coordinates)

denoting φ as the angle relative to the z axis and ρ as the distance to the origin

ρ = √(x²+y²+z²) = √(3+9+4) = 4

z = ρ*cos φ →  φ = cos⁻¹ (z/ρ) =cos⁻¹ (1/2) = π/3

then (ρ, θ, φ) = (4 , π/3, π/3) (spherical coordinates)

Place values after decimal point:

The first digit after the decimal point is the tenth place, the second digit after the decimal point is the hundredths place and the third is the thousandth place.

When you are rounding off to a place value, look at the digit in the place value before that. If the previous digit is 5 and above, increase the value of the digit by 1.

Q22. tenth

The tenth place has been rounded up from '3' to '4' since the hundredth place is a 7 which is greater than 5.

Q23. hundreds

The digit '4' is in the hundreds place. Since the tens place is 2, the number is rounded off to 400 instead of 500.

Q24. ones

The ones place have been rounded up to a digit 9 since the tenth place is greater then 5.

Q25. hundredth

The digit 5 is in the hundredth place

Q26. tens

Q27. thousandth

A general guideline for these few questions is to look at the number that has been rounded off and identify the place value of the last digit written that is not 0. This digit has also been changed if the digit in the next place value is greater than 5.

Let's look at Q26. The last digit of the rounded off number which is not zero is the digit '5' in the tens place. Thus, the number has been rounded off to the nearest tens.