Question

13. 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?

Answer 1

Since the Power is directly proportional to the velocity, their ratio $\frac{P}{v^{3} }$ is constant.

Therefore $\frac{500}{ 10^{3} } = \frac{x}{ 12^{3} }$ which is a proportion:

500 : 1000 = x : 1728

that can be easily solved:

x = (500×1728)/1000 = 864kW

Answer 2

Let's assume

The power  (in kilowatts) generated by a wind turbine as P

the wind speed  (in meters per second) as v

we are given

The power P (in kilowatts) generated by a wind turbine varies directly as the cube of the wind speed v (in meters per second)

so, we get

$P=kv^3$

where

k is a proportionality constant

now, we are given

If a turbine generates 500kW in a 10m/s wind

so, at v=10m/s , P=500kW

now, we can use it and find k

$500=k(10)^3$

$k=0.5$

now, we can plug it back

and we get

$P=0.5v^3$

we can plug v=12m/s

and find P

$P=0.5(12)^3$

$P=864kW$

so,

The power  generates in a 12m/s wind is 864kW........Answer