How do you find the gauge pressure if all you are given is that the hose will spray to a height of 10 meters?
Apply Bernoulli's Equation.
Before it hits the nozzle the water isn't moving very fast, so you can consider its speed negligible.
At the 10 meter height its velocity is zero.
You know the change in altitude.
So you can use Bernoulli's equation to find the difference in the two pressures.
If a material has a density of 8.7kg/liter and 2kg of the material is suspended from a string and immersed in water, how do you find the tension in the string. I can find how much volume if I divide the mass by the density and I can find the gravitational force but from those how do you find the tension?
The weight is pulled down by gravity, pulled up by the tension in the string and the buoyant force. It's in equilibrium, so the sum of all these forces is zero.
How do you find the rate of volume flow from a faucet when you have the cross-sectional diameter which is .6cm and the pressure head which is 400000 kPa?
From the pressure head and Bernoulli's equation you find the speed of the exiting water; again you assume that the squared speed of the water before it reaches the faucet opening is negligible.