Respuesta :
Explanation:
It is given that,
[tex]\frac{dx}{dt}[/tex] = -40 mi/h, [tex]\frac{dx}{dt}[/tex] = -40 mi/h
The negative sign indicates that x and y are decreasing.
We have to find [tex]\frac{dz}{dt}[/tex]. Equation for the given variables according to the Pythagoras theorem is as follows.
[tex]z^{2} = x^{2} + y^{2}[/tex]
Now, we will differentiate each side w.r.t 't' as follows.
[tex]2z\frac{dz}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}[/tex]
or, [tex]\frac{dz}{dt} = \frac{1}{z}(x\frac{dx}{dt} + y\frac{dy}{dt})[/tex]
So, when x = 4 mi, and y = 3 mi then z = 5 mi.
As, [tex]\frac{dz}{dt} = \frac{1}{z}(x\frac{dx}{dt} + y\frac{dy}{dt})[/tex]
= [tex]\frac{1}{5}(4 \times (-40) + 3 \times (-40))[/tex]
= [tex]\frac{-140 - 120}{5}[/tex]
= 52
Thus, we can conclude that the cars are approaching at a rate of 52 mi/h.
