A15 m long garden hose han an inner diameter of 2.5 cm. one end is connected to a spigot; 20 c water flows from the other end at a rate of 1.2 l/s. what is the gauge pressure at the spigot end of the hose

We can solve this problem using Hagen–Poiseuille equation. Derivation of this equation is a bit complicated so I will just write down the equation.

This equation gives you the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of the constant cross section.
L is the length of the cylinder, Q is the volumetric flow rate, R is the radius of the pipe, and is dynamic viscosity.
Dynamic viscosity of water at 20 Celsius is 0.001 PaS. 
Now we can calculate the pressure drop:

Pe = 1/2k(x)^2  and .05n/m = .05x10^-9m = x