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 500kW in a 10m/s wind, how much power does it generate in a 12m/s wind.

To find out how much power the turbine generates in a 12 m/s wind, we need to use the direct variation relationship between power and wind speed.

The problem states that the power generated by the wind turbine varies directly as the cube of the wind speed. Mathematically, we can express this relationship as:

P ∝ v^3

where P represents the power generated and v represents the wind speed.

To solve the problem, we can set up a proportion by using the power and wind speed values given in the problem:

P1 / v1^3 = P2 / v2^3

Let's plug in the values given in the problem:

P1 = 500 kW (power generated in a 10 m/s wind)
v1 = 10 m/s (wind speed)

We need to find P2, the power generated in a 12 m/s wind.

v2 = 12 m/s (wind speed)

Now we can substitute these values into our proportion:

500 / 10^3 = P2 / 12^3

Simplifying, we get:

500 / 1000 = P2 / 1728

0.5 = P2 / 1728

To solve for P2, we can cross-multiply:

0.5 * 1728 = P2

864 = P2

Therefore, the turbine generates 864 kW of power in a 12 m/s wind.

864