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?
We can solve this problem using the formula for direct variation:
P = kv³
where P is the power generated by the wind turbine and v is the wind speed.
To find the value of k, we can use the given information that the turbine generates 500 kW in a 10 m/s wind. Plugging these values into the formula, we have:
500 = k(10³)
Simplifying, we get:
500 = 1000k
Dividing both sides by 1000, we find that k = 0.5.
Now we can use this value of k to find the power generated by the turbine in a 12 m/s wind:
P = 0.5(12³)
Simplifying, we have:
P = 0.5(1728)
P = 864
Therefore, the turbine generates 864 kW in a 12 m/s wind.