Mathematics

Problem

## Chapter Opener Problem (from page 735) Problem: For an efficient windmill, the power generated in watts is given by the equation

P=k r^{2} v^{3}

where

r

is the radius of the propeller in meters,

v

is the wind velocity in meters per second, and

k

is a constant with units of

\mathrm{kg} / \mathrm{m}^{3}

. The exact value of

k

depends on various characteristics of the windmill. Suppose a windmill has a propeller with radius 5 meters and

k=0.134 \mathrm{~kg} / \mathrm{m}^{3}

. (a) Find the function

P(v)

, which gives power as a function of wind velocity. (b) Find

P^{\prime}(7)

, the rate of change in power generated with respect to wind velocity, when the wind velocity is 7 meters per second.