Mathematics

Problem

For a certain moving point, the coordinate

x

depends on time

t

according to

x=\sqrt{4t+1}

. A second quantity is defined by

y=(3x^{2}+2x)^{\tfrac{5}{2}}

. Determine the rate of change

\dfrac{dy}{dt}

when

t=3\,\mathrm{s}

. a. (i) Obtain

\dfrac{dy}{dx}

in its simplest form. b. (ii) Find

\dfrac{dx}{dt}

in terms of

t

. c. (iii) Hence compute the numerical value of

\dfrac{dy}{dt}

t=3\,\mathrm{s}

, writing the final answer with appropriate units.