A point moves along the curve $3 y + y ^ { 2 } - 8 x = 2$. When the point is at $\left( - \frac { 1 } { 2 } , - 1 \right)$, its $x$-coordinate is increasing at the rate of 3 units per second. How fast is its $y$-coordinate changing at that instant of time?
A) 24 units/sec
B) 26 units $/ \mathrm { sec }$
C) $- 24 \mathrm { units } / \mathrm { sec }$
D) $- 22$ units $/ \mathrm { sec }$

Question

Quizplus · Accepted Answer

Implicit differentiation gives $$3+2yy'-8=0 \quad \Rightarrow \quad y'=\frac{4}{2y-3}$$At the given point, $y=-1$, so $$y'=\frac{4}{2(-1)-3}=\frac{4}{-5}=-\frac{4}{5}$$Since $x$ is increasing at the rate of 3 units per second, we have $$\frac{dx}{dt}=3$$The Chain Rule gives$$\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}=y'\frac{dx}{dt}=\left(-\frac{4}{5}\right)(3)=-\frac{12}{5}$$Therefore, the $y$-coordinate is changing at the rate of $-12/5$ units per second, which is equivalent to $-24$ units per second.

A Point Moves Along the Curve 3y+y2−8x=23 y + y ^ { 2 } - 8 x = 23y+y2−8x=2

A Point Moves Along the Curve $3 y + y ^ { 2 } - 8 x = 2$