Consider a river flowing toward a lake at an aver- age velocity of 3 m/s at a rate of $500 m^3/s$ at a location 90 m above the lake surface. Determine the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location.

Assume that the double integral of a positive function f extended over a region S reduces to the given repeated integral. In each case, make a sketch of the region S and interchange the order of integration. $\int_{-1}^{1}\left[\int_{-\sqrt{1-x^{2}}}^{1-x^{2}} f(x, y) d y\right] d x$

Consider a river flowing toward a lake at an average speed of $3 \mathrm{~m} / \mathrm{s}$ at a rate of $500 \mathrm{~m}^3 / \mathrm{s}$ at a location $55 \mathrm{~m}$ above the lake surface. Find the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location.

A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

A stream flowing into a lake brings with it a pollutant at a rate of $8$ metric tons per year. The river leaving the lake removes the pollutant at a rate proportional to the quantity in the lake, with constant of proportionality $-0.16$ if time is measured in years.

(b) What is the quantity of pollutant in the lake after a long time?

An oil pump is drawing $35 \mathrm{~kW}$ of electric power while pumping oil with $\rho=860 \mathrm{~kg} / \mathrm{m}^3$ at a rate of $0.1 \mathrm{~m}^3 / \mathrm{s}$. The inlet and outlet diameters of the pipe are $8 \mathrm{~cm}$ and 12 $\mathrm{cm}$, respectively. If the pressure rise of oil in the pump is measured to be $400\ \mathrm{kPa}$ and the motor efficiency is $90$ percent, find the mechanical efficiency of the pump.

Electric power is to be generated by installing a hydraulic turbine–generator at a site 110 m below the free surface of a large water reservoir that can supply water steadily at a rate of 900 kg/s. If the mechanical power output of the turbine is 800 kW and the electric power generation is 750 kW, determine the turbine efficiency and the combined turbine–generator efficiency of this plant. Neglect losses in the pipes.

Consider a river flowing toward a lake at an average speed of 4 m/s at a rate of $500 m^3/s$ at a location 70 m above the lake surface. Determine the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location.

Find a power series representation of f. Use a geometric series and properties of a power series. $f(x)=\ln (1-2 x)$

Consider a hollow-core printed circuit board 9 cm high and 18 cm long, dissipating a total of 15 W. The width of the air gap in the middle of the PCB is 0.25 cm. If the cooling air enters the 12-cm-wide core at $25^\circ C$ and 1 atm at a rate of 0.8 L/s, determine the average temperature at which the air leaves the hollow core.

An oil pump is drawing 44 kW of electric power while pumping oil with $\rho$= 860 kg/m^3 at a rate of 0.1 m^3/s. The inlet and outlet diameters of the piper are 8 cm and 12 cm, respectively. If the pressure rise of oil in the pump is measured to be 500 kPa and the motor efficiency is 90 percent, determine the mechanical efficiency of the pump.

Consider a river flowing toward a lake at an average velocity of 3 m / s at a rate of 500 m $^3 / s$ at a location 90 m above the lake surface. Determine the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location.

Electric power is to be generated by installing a hydraulic turbine-generator at a site 120 m below the free surface of a large water reservoir that can supply water at a rate of 1500 kg/s steadily. Determine the power generation potential.

A water jet that leaves a nozzle at 60 m/s at a flow rate of 120 kg/s is to be used to generate power by striking the buckets located on the perimeter of a wheel. Determine the power generation potential of this water jet.