Final answer:
The rate of change of y with respect to time, {d y}{d t}, when x=3 and y=-1 is found to be 3 m/s after differentiating the given equation implicitly with respect to time and substituting the given values.3 m/s.
Step-by-step explanation:
The student is given the equation x² + 5 y² + 6 y = 8 and is required to find {d y}{d t} given that {d x}{d t} = 2 when x = 3 and y = -1. To solve this problem, we must first implicitly differentiate the equation with respect to time t, then we substitute the values of x, y, and {d x}{d t} to solve for {d y}{d t}.
Taking the derivative with respect to time:
2x{d x}{d t} + 10y{d y}{d t} + 6{d y}{d t} = 0
Substituting the known values:
2(3)(2) + 10(-1){d y}{d t} + 6{d y}{d t} = 0,
12 - 10{d y}{d t} + 6{d y}{d t} = 0,
-4{d y}{d t} = -12,
{d y}{d t} = ⅓. So, the rate of change of y with respect to time when x = 3 and y = -1 is 3 m/s.