Answer
a. $\frac{dx}{dt} = \frac{-\sqrt 5}{4}$
b. $\frac{dy}{dt} = \frac{4\sqrt 5}{5}$
Work Step by Step
a. $\frac{d(4x^2 + 9y^2)}{dt} = \frac{d(36)}{dt}$
$8x \times \frac{dx}{dt} + 18y \times \frac{dy}{dt} = 0$
$8(2) \times \frac{dx}{dt} + 18(\frac{2\sqrt 5}{3}) \times \frac{dy}{dt} = 0$
$16 \times \frac{dx}{dt} + 18(\frac{2\sqrt 5}{3}) \times \frac{1}{3} = 0$
$16 \times \frac{dx}{dt} + 4\sqrt 5 = 0$
Solve for $\frac{dx}{dt}$:
$16 \times \frac{dx}{dt} = -4\sqrt 5$
$\frac{dx}{dt} = \frac{-4\sqrt 5}{16}$
$\frac{dx}{dt} = \frac{-\sqrt 5}{4}$
b. $8x \times \frac{dx}{dt} + 18y \times \frac{dy}{dt} = 0$
$8(-2)\frac{dx}{dt} + 18(\frac{2\sqrt 5}{3}) \times \frac{dy}{dt} = 0$
$-16(3) + 12\sqrt 5 \times \frac{dy}{dt} = 0$
$-48 + 12\sqrt 5 \times \frac{dy}{dt} = 0$
Solve for $\frac{dy}{dt}$
$\frac{dy}{dt} = \frac{48}{12\sqrt 5}$
$\frac{dy}{dt} = \frac{4}{\sqrt 5}$
$\frac{dy}{dt} = \frac{4\sqrt 5}{5}$
