Final answer:
To find the rate of change of z with respect to time, we differentiate the equation z^2 = x^3 + y^2 implicitly with respect to time and substitute the given values to solve for dz/dt. At the point (4, 0), dz/dt is found to be -6 units of z per unit of time.
Step-by-step explanation:
The student is asking to find the rate of change of z with respect to time, given that z is a function of x and y, which are themselves functions of time (t). The provided equation is z2 = x3 + y2. We must use implicit differentiation with respect to time to find dz/dt at the given point (x, y) = (4, 0).
From the given data, we have dx/dt = -2 and dy/dt = -3, and z > 0. Differentiating both sides of the equation z2 = x3 + y2 with respect to time, we get:
2z(dz/dt) = 3x2(dx/dt) + 2y(dy/dt).
At (x, y) = (4, 0), we substitute the given values: x = 4, y = 0 , dx/dt = -2, and dy/dt = -3 into the differentiated equation and solve for dz/dt:
2z(dz/dt) = 3(4)2(-2) + 2(0)(-3).
2z(dz/dt) = 3(16)(-2) + 0,
2z(dz/dt) = -96.
Given z is positive and z2 = 43 + 0 = 64 at the point (4, 0), we find z to be 8 (since z > 0).
Thus,
2(8)(dz/dt) = -96,
16(dz/dt) = -96,
dz/dt = -96/16,
dz/dt = -6.
Hence, the rate of change of z with respect to time at the point (4, 0) is -6 units of z per unit of time.