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Wind farms are a source of renewable energy found around the world. The power P (in kilowatts) generated by a wind turbine varies directly as the cube of the wind
speed v (in meters per second). If a turbine generates 500 kW in a 10 m/s wind, how much power does it generate in a 12 m/s wind?
Step 1: Write the given information in the form of an equation.
The power P (in kilowatts) generated by a wind turbine varies directly as the cube of the wind speed v (in meters per second).
P = k * v^3
where k is a constant of proportionality.
Step 2: Use the given information to find the value of the constant of proportionality k.
In a 10 m/s wind, the turbine generates 500 kW.
Substituting these values into the equation:
500 = k * 10^3
Simplifying:
k = 500 / (10^3)
k = 0.05
Step 3: Use the value of k to find the power generated in a 12 m/s wind.
Substituting the value of k and the wind speed v = 12 into the equation:
P = 0.05 * 12^3
P = 0.05 * 1728
P = 86.4
Answer: The turbine generates 86.4 kW in a 12 m/s wind.