A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 10 - 5(x² + y² + z²) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K s Vwds. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m·K)). (Use symbolic notation and fractions where needed.) VwdS= -K Vu kW

Please do this carefully.

Calculate the rate of flow of waste thermal energy out of the engine in the exhaust gases in kW (to 1 decimal point). If the air inlet temperature is 35 °C, the air to fuel ratio is 10:1, the inlet air mass flow rate is 37 kg/hr, the specific heat capacity of exhaust gas is 1.13 kJ/kg/K, and the exhaust temperature is 500 °C. Add your answer Integer, decimal or E notation allowed

An electric hot water heater consumes 3.1 kilowatts of electricity and converts it to heat. How long will it take the water heater to heat a 67 gallon tank of water from 10 degrees Celsius to 50 degrees Celsius? (1 kilogram of water is 0.37 gallons, 1 Calorie = 4200 J).  It may be helpful to refer back to the weekly handout for guidance on this problem. Your final answer should be in minutes (rounded to the nearest 10 minutes).

Tire manufacturers recommend that you check your tire pressure monthly to avoid unnecessary tire wear and enhance safety. This is best done when the tires are cold meaning; the car has not been driven for at least 2 hours. Since you leave for work before the sun comes up you decide to inflate your tires to the recommended pressure right after you get home from work in the afternoon when the tires are still warm. What would you find if you rechecked your tire pressure after the tires have cooled down? A.Tire pressure has decreased because there is a decrease in kinetic energy of themolecules resulting in more collisions with the inside walls of the tire thusincreasing the pressure inside the tire. B.Tire pressure has increased because there is a decrease in kinetic energy of the molecules resulting in fewer collisions with the inside walls of the tire thus decreasing the pressure inside the tire. C.Tire pressure has increased because there is an increase in kinetic energy of the…