Find the point `(alpha,)beta` on the ellipse `4x^2+3y^2=12 ,` in the first quadrant, so that the area enclosed by the lines `y=x ,y=beta,x=alpha` , and the x-axis is maximum.

Explanation

Let the speed of the motor boat be v m/h.Then
Velocity of the boat relative ot the current =(v-c) m/h
If s miles is the distacne covered then the time taken to cover this distance is t=s/(v-c)hours.
Since the petrol burnt =`kv^(3)` per hour where k is conatant ,the totla amount of petrol burnt for a distance of s miles,
z=`kv^(3) s/(v-c)`
`(dz)/(dv)=2kSv^(2)(v-3c//2)/(v-c)^(2)`
For maximum or minimum of z ,dz/dz =0 or v =`3c//2`
If v is little less or little greater than `3c//2` then the sign of dz/dv changes from -ve to + ve .Hence ,z is minimum when v=`3c//2 m/h`.
Since minima is the only extreme value, z is least at v=`3c//2` , i.e the most economical speed is `3c//2 m//h`