The potential difference VVV, in millivolts, across a resistor in an alternating current circuit is modeled by the function V(t)=7cost−24sintV(t) = 7\cos t - 24\sin tV(t)=7cost−24sint, where t t\,t is the time in milliseconds since the circuit was closed.
Express V(t)V(t)V(t) in the form Rcos(t+α)R\cos(t + \alpha)Rcos(t+α), where R>0 R > 0\,R>0 and 0<α<π2\displaystyle 0 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of R R\,R and the value of α\alphaα, in radians, to 3 decimal places.
The curve with equation y=costy = \cos ty=cost is transformed onto the curve with equation y=V(t)y = V(t)y=V(t) by a sequence of two transformations.
Given that the first transformation is a stretch and the second is a translation:
(i) Describe fully the transformation that is a stretch. (ii) Describe fully the transformation that is a translation.
The power efficiency G G\,G of a specific component is given by G(t)=210075+(V(t))2\displaystyle G(t) = \frac{2100}{75 + (V(t))^2}G(t)=75+(V(t))22100. Find the range of GGG.