Question

The tip speed ratio of a turbine is the ratio R = T/W, where T is the speed of the tip of a blade and W is the speed of the wind. (Engineers have found empirically that a turbine with n blades extracts maximum power from the wind when R = 2π/n.) Calculate dR/dt (t in minutes) if W = 75 km/h and W decreases at a rate of 4 km/h per minute, and the tip speed has constant value T = 150 km/h.

Answer 1

`\displaystyle(dR)/(dt) \approx 0.107`

Explanation:

We are given the following information in the question:

`R = \displaystyle(T)/(W)`

where, R is the tip speed, T is the speed tip of blade and W is the speed of wind.

We have to find
`(dR)/(dt)`, whem W = 75 km per hour,
`(dW)/(dt) = -4\text{ km per hour}`, T = 150 km per hour

Formula:

`\displaystyle(dR)/(dt) = \displaystyle(dR)/(dW)\displaystyle(dW)/(dt) + \displaystyle(dR)/(dT)\displaystyle(dT)/(dt)\\\\= \displaystyle(dR)/(dW)\displaystyle(dW)/(dt)\\\\\text{ as T is constant with respect to t}\\\\= -\displaystyle(T)/(W^2)\displaystyle(dw)/(dT)`

Putting all the value, we have:

`\displaystyle(dR)/(dt) = -\displaystyle(150* -4)/(75* 75) = 0.106666666667 \approx 0.107`