To determine the period of the function [tex]\( y = \frac{3}{2} \cot \left( \frac{3}{5} x \right) + 5 \)[/tex]:
1. Identify the general form and parameters:
The given function is [tex]\( y = A \cot(B x) + C \)[/tex].
Here, [tex]\( A = \frac{3}{2} \)[/tex], [tex]\( B = \frac{3}{5} \)[/tex], and [tex]\( C = 5 \)[/tex].
2. Understand the period of the basic function [tex]\(\cot(x)\)[/tex]:
The period of the basic cotangent function [tex]\( \cot(x) \)[/tex] is [tex]\( \pi \)[/tex].
3. Adjust the period based on the coefficient [tex]\( B \)[/tex]:
For the function [tex]\( \cot(B x) \)[/tex], the period is given by [tex]\( \frac{\pi}{|B|} \)[/tex].
4. Calculate the period:
- [tex]\( B = \frac{3}{5} \)[/tex]
- Therefore, the period is [tex]\( \frac{\pi}{\frac{3}{5}} \)[/tex].
5. Simplify the expression:
- [tex]\( \frac{\pi}{\frac{3}{5}} = \pi \times \frac{5}{3} = \frac{5\pi}{3} \)[/tex].
Thus, the period of the function [tex]\( y = \frac{3}{2} \cot \left( \frac{3}{5} x \right) + 5 \)[/tex] is [tex]\( \frac{5\pi}{3} \)[/tex] units.
When comparing this result to the given options:
- [tex]\( \frac{\pi}{5} \)[/tex] units
- [tex]\( \frac{3\pi}{5} \)[/tex] units
- [tex]\( \frac{2\pi}{3} \)[/tex] units
- [tex]\( \frac{5\pi}{3} \)[/tex] units
The correct choice is [tex]\( \frac{5\pi}{3} \)[/tex] units, which corresponds to the last option.
