Question

Two spacecraft are following paths in space given by r1=〈sin⁡(t), t, t²〉 and r²=〈cos⁡(t), 1−t, t³〉. If the temperature for the points is given by T(x, y, z)=x²y(9−z), use the Chain Rule to determine the rate of change of the difference D in the temperatures the two spacecraft experience at time t=2.

Answer 1

To find the rate of change of the temperature difference at time t=2 for the two spacecraft, we can use the Chain Rule to calculate the partial derivatives of the temperature function and the derivatives of the position vectors with respect to time. By substituting the values and evaluating the derivatives, we can determine the rate of change.

Step-by-step explanation:

To determine the rate of change of the difference in temperatures that the two spacecraft experience at time t=2, we need to use the Chain Rule. First, we find the partial derivatives of the temperature function T(x, y, z) = x²y(9-z) with respect to x, y, and z:

1. ∂T/∂x = 2xy(9-z)
2. ∂T/∂y = x²(9-z)
3. ∂T/∂z = -x²y

Next, we find the derivative of the position vectors r₁ and r₂ with respect to t:

• r₁'(t) = 〈cos(t), 1, 2t〉
• r₂'(t) = 〈-sin(t), -1, 3t²〉

Using the Chain Rule, we can calculate the rate of change of the temperature difference at t=2 by substituting the values into the above equations and evaluating the derivatives:

1. ∂T/∂t = ∂T/∂x * ∂x/∂t + ∂T/∂y * ∂y/∂t + ∂T/∂z * ∂z/∂t
2. Plug in the values for ∂T/∂x, ∂T/∂y, ∂T/∂z, ∂x/∂t, ∂y/∂t, and ∂z/∂t
3. Evaluate the expression at t=2

This will give you the rate of change of the temperature difference at time t=2 for the two spacecraft.