Exercise 15

Suppose that the temperature on a metal plate is given by the function $T$ with $$\begin{gathered} T(x,y) = 100-(x^2 + 4y^2), \end{gathered}$$ where the temperature is measured in degrees Fahrenheit and $x$ and $y$ are each measured in feet. Now suppose that an ant is walking on the metal plate in such a way that it walks in a straight line from the point $(1,4)$ to the point $(5,6)$.

1. Find parametric equations $(x(t),y(t))$ for the ant's coordinates as it walks the line from $(1,4)$ to $(5,6)$.

2. What can you say about $\frac{dx}{dt}$ and $\frac{dy}{dt}$ for every value of $t$?

3. Determine the instantaneous rate of change in temperature with respect to $t$ that the ant is experiencing at the moment it is halfway from $(1,4)$ to $(5,6)$, using your parametric equations for $x$ and $y$. Include units on your answer.