calculate the pressure needed for the pump of a 3.0 cm radius firehouse to pump water through its nozzle of radius 2.0 cm at an average speed of 4.0 m/s when the nozzle is 15 m above the fire truck.

Answer 1

Firehouse or firehose?
How long is the hose?
They probably want you to neglect the pressure drop in the hose.
The average speed of the water in the hose ahead of the nozzle will be
v = 4.0 m/s*(4/9) = 16/9 m/s.
The continuity equation has been used.

Now use the Bernoulli equation.

P = Po + (rho)*g*H + (1/2)*(rho)*v^2
Rho is the density of water; g is the acceleration of gravity; H = 15 m. Po is atmospheric pressure.

What they really want is P - Po, the gauge pressure.