To calculate the gauge pressure at a depth of 10m in a muddy lake, the density variation is given by ρ=ρ0+kh, where ρ0 is 1000 kg/m^3 and k is 100 kg/m^4. Using the formula for pressure, P = ρ × g × h, and disregarding atmospheric pressure for gauge pressure, the density at 10m is calculated to be 2000 kg/m^3. The gauge pressure is then determined by multiplying this density by the acceleration due to gravity and the depth. Additionally, the dryness fraction of steam after throttling from 14 bar to 1.1 bar and reaching a temperature of 110°C can be calculated using steam tables.