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Wind farms are a source of renewable energy found around the world. The power P(in kilowatts)...

Question:

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 700kW in a 5m.s wind, how much power does it generate in a 14m/s wind?

Ratio and Proportion

There are numerous known applications of ratio and proportion in real life. Ratio is basically the relation between two quantities and it is usually represented in fraction or scale. Moreover, proportion is the relation of two ratios.

Answer and Explanation:

If the power P varies directly with the cube of the wind speed v, thus

{eq}P=kv^3 {/eq}

where k is the proportionality constant. By rearranging the above equation, we have

{eq}k=\dfrac{P}{v^3} {/eq}

Then, substitute the given values, which are shown below.

{eq}P_1=700\,\rm kW\\ v_1=5\,\rm \dfrac{\rm m}{\rm s}\\ v_2=14\,\rm \dfrac{\rm m}{\rm s}\\ {/eq}

Therefore,

{eq}k=\dfrac{700\,\rm kW}{\left(5\,\rm \dfrac{\rm m}{\rm s}\right)^3}=\dfrac{P_2}{\left(14\,\rm \dfrac{\rm m}{\rm s}\right)^3}\\[0.2cm] P_2=\dfrac{700\,\rm kW}{\left(5\,\rm \dfrac{\rm m}{\rm s}\right)^3}\left(14\,\rm \dfrac{\rm m}{\rm s}\right)^3\\[0.2cm] P_2=\dfrac{700\,\rm kW}{125}(2744)\\[0.2cm] \color{blue}{P_2=15366.4\,\rm kW\,\,\,\rm or\,\,\,\approx 15.37\,\rm MW}\\[0.2cm] {/eq}


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What is Ratio in Math? - Definition & Overview

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