Question

Wind power P from a turbine varies directly as the square of the length r of one ofits blades. Two common blade lengths for commercial wind turbines are 35m and 50m. When the blade length is 35m about 1.5 megawatt of power is produced under favorable conditions. How much power would be produced, under favorable conditions, by a turbine with 50m blades.

Answer 1

Explanation:

If two variables are directly proportional, it means that an increase in the value of one variable would cause a corresponding increase in the other variable. Also, a decrease in the value of one variable would cause a corresponding decrease in the other variable.

Given that P varies directly with r², if we introduce a constant of proportionality, k, the expression becomes

P = kr²

If P = 1.5 when r = 35, then

1.5 = k × 35²

k = 1.5/35² = 1.5/1225

Therefore, the direct variation function is

P = 1.5r²/1225

When r = 50, then

P = 1.5 × 50²/1225

P = 3.06 megawatt

Answer 2

3.06 megawatts

Explanation:

Firstly, we write the proportionality equation.

wind power varies directly as the square of the length r of one of the blades;

P ∝ r^2

Let’s introduce a constant of proportionality k; This means;

P = kr^2

Now let’s calculate the value of k when 1.5 megawatts and r is 35m

1.5 mw = 35^2 * k

k = 1.5/35^2 = 1.5/1225 = 0.001224489796 MW/m^2

Now we want to calculate the amount of megawatts to be produced by a turbine with 50m blades.

That would be;

P = 0.001224489796 * 50^2 = 3.06 megawatts