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) $- 24$ units/sec
D) $- 22$ units/sec

Question

Quizplus · Accepted Answer

Explanation:To find the rate of change of $y$, we differentiate both sides of the given equation with respect to time $t$:$$\frac{d}{dt}(3y + y^2 - 8x) = \frac{d}{dt}(2)$$$$3\frac{dy}{dt} + 2y\frac{dy}{dt} - 8\frac{dx}{dt} = 0$$Substituting the given values, we get:$$3\frac{dy}{dt} + 2(-1)\frac{dy}{dt} - 8(3) = 0$$$$3\frac{dy}{dt} - 2\frac{dy}{dt} - 24 = 0$$$$\frac{dy}{dt} = 24$$Therefore, the $y$-coordinate is changing at a rate of 24 units/sec.

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 $3 y + y ^ { 2 } - 8 x = 2$