Ahiker is hiking in a valley. the height of the valley is h(x, y)=4x2+y2 where x and y are the east-west and north-south distances from the valley floor respectively. the hiker follows an elliptical path x(t)=2 cos(t), y(t)=4 sin(t) around the valley. compute the time derivative of the height in two ways. first expand the composite function h(x(t),y(t)) explicitly in terms of t. second use the chain rule.

A.

Step-by-step explanation:

A. The problems asked for 2 ways to solve it, expanding the equation with the substitution  x(t)=2 cos(t) and y(t)=4 sin(t) to differentiate it . The other way is by chain rule.

Expanding and differentiating:

We start by substituting x(t)=2 cos(t) and y(t)=4 sin(t) in h(x,y)=4x2+y2:

So, in the path that the hiker chose:

Chain rule:

We start differentiating h(x,y) using chain rule as follows:

Now, it´s easy to find all these derivatives:

Now we replace them in the chain rule, with the replacement x=2cos(t) and y=4sin(t) in the x,y that are left and we operate everything:

This will be our answer

4 + 5 7/9= 9 7/9

step-by-step explanation: