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 ms/
wind, how much power does it generate in a 12 ms/
wind?
(2 points)
To solve this problem, we can set up a proportion using the formula for direct variation.
We know that the power P generated by a wind turbine varies directly as the cube of the wind speed v. So, we can write the proportion as:
P/500 = (v/10)^3
To find the power generated in a 12 m/s wind, we can substitute the value of wind speed (v) into the proportion and solve for P.
P/500 = (12/10)^3
P/500 = (1.2)^3
P/500 = 1.728
Cross multiplying gives us:
P = 1.728 * 500
P = 864
Therefore, the turbine generates 864 kW in a 12 m/s wind speed.
To solve this problem, we can use the concept of direct variation.
Let's denote the power generated by the wind turbine in a 10 m/s wind as P1 and the power generated in a 12 m/s wind as P2.
According to the problem, the power generated by the wind turbine varies directly as the cube of the wind speed. Mathematically, we can write this as:
P1 ∝ v^3
Using this relationship, we can write the equation:
P1 = k * v^3
where k is the constant of variation.
Now, we can plug in the given values for P1 and v1 to find the value of k:
500 = k * (10)^3
Simplifying the equation:
500 = 1000k
Dividing both sides of the equation by 1000:
k = 0.5
Now that we have the value of k, we can use it to find P2. Substituting the values of k and v2 into the equation, we have:
P2 = 0.5 * (12)^3
Calculating the value:
P2 = 0.5 * 1728
P2 = 864
Therefore, the wind turbine will generate 864 kW of power in a 12 m/s wind.
To solve this problem, we need to use the direct variation formula, which states that if a quantity varies directly with another quantity, it can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.
In this case, the power P (in kW) is the dependent variable, and the wind speed v (in m/s) is the independent variable. The equation would be written as P = kv^3.
We are given that the turbine generates 500 kW in a 10 m/s wind. So we can substitute these values into the equation:
500 = k * 10^3
500 = k * 1000
k = 500/1000
k = 0.5
Now we have the constant of variation, k. We can use this value to find the power generated in a 12 m/s wind.
P = 0.5 * 12^3
P = 0.5 * 1728
P = 864
Therefore, the turbine generates 864 kW in a 12 m/s wind.