Answer :
Answer:
Step-by-step explanation:
We are given the following:
[tex]R = \{(x,y)|0\leq x \leq 1, 0 \leq y \leq 1 \}[/tex]
and [tex]T(x,y) = 100-25x - 40 y[/tex]
a). Recall that a level curve of a function f(x,y) is given by [tex]R_c = \{(x,y) | f(x,y) = c\}[/tex] where c is a constant. That is, all the points in the set of interest to which the function applied to the points is exactly the value c.
Consider c = 80. So we get
[tex] 100-25x-40y = 80[/tex]
which implies that [tex] y = \frac{-25}{40}x+0.5[/tex](Graph 1).
We can also consider c=60, which gives us
[tex] 100-25x-40y = 60[/tex]
which implies that [tex]y = \frac{-25}{40}x+1[/tex]. (Graph 2)
b)Recall that the gradient of a function f(x,y) is given by
[tex]\nabla f = (\frac{df}{dx}, \frac{df}{dy})[/tex]
In this case,
[tex]\frac{dT}{dx} = -25, \frac{dT}{dy} = -40[/tex]
Thus, the gradient of T is given by
[tex]\nabla T =(-25,-40)[/tex]
