Answer:
Hence the distance between automobile and farmhouse is increase at rate of 33.541 mph.
Step-by-step explanation:
Given:
farmhouse distance from highway is 2 miles
An automobile traveling at 75 mph.
To Find:
Distance between automobile and farmhouse when automobile pass 4 miles way from intersection of highway.(rate miles per hr)
Solution:
The solution is required in rate ,(Refer the attachment)
so using derivative with respect to time
Consider Triangle ABC,
<em>AB=2 miles</em>
<em>x=BC=X km distance from highway intersection</em>
<em>y=AC= Distance between car and the farmhouse.</em>
Using Pythagoras Theorem we get ,
y^2=x^2+2^2
y^2=x^2+4
y=Sqrt of{(x^2+4)}
Differentiate w.r.t 't' we get
2y*(dy/dt)=2x*(dx/dt)
dy/dt=(x/y)*(dx/dt)
dy/dt=(x/Sqrt of{(x^2+4)})*(dx/dt)............(here x=4 miles)
dy/dt=(2/Sqrt(20))*(dx/dt)
here dx/dt is the rate of change of distance i.e. speed=75
dy/dt=(2/Sqrt(20))*(75)
=0.44721*75
=33.541 mph