As an engineering student $,$ with a foundation in Fluid Mechanics, you have

observed a notable decrease in water flow through the showerhead when others use the toilet.

Intrigued by this phenomenon, you have initiated a small investigation to delve deeper into the

matter. Upon inspecting the plumbing system in your bathroom, you have discovered that both

the outlet of the showerhead and the toilet reservoir in your house are connected using $1\mathrm{.}5-$cm

diameter copper pipes with threaded connectors. Additionally, you have identified the presence

of a valve, elbow, tee, and branch attached to the $12$m plumbing system.

During your investigation, you have also determined that the inlet pressure reads $100$ kPa $($gage$)$

and water flow at $2\mathrm{.}98$ m$/$s during showering when the toilet reservoir is full. However, once the

toilet flushed the velocity reduced to $2\mathrm{.}377$ m$/$s$.$ Now, armed with these findings, you are keen to

explore the ways to maintain the flow rate of the shower. Your objectives include:

$1.$ Develop a schematic diagram of the plumbing system, highlighting key components and

connections.

$2.$ Perform calculations to identify the effects of toilet flushing on showerhead head loss and

flow rate.

$3.$ Present findings and recommendations for optimizing the plumbing system to ensure

consistent water flow.