(a) At what rate is the radius of a spherical balloon changing at the time that the radius is 10cm, if air is being pumped in at a rate of 40 cm3s¡1?
(b) Find an expression for dy/dx
given that 3xy + ey = 3.
1) The volume of a sphere is . We know that the volume and radius are functions of time here, hence, using implicit differentiation:
Hence:
You know what is, and you know what the radius is at our point of interest. So plug them in!
2)
You could take out a factor of y on the LHS, and then move all other terms to the RHS, giving you a nice explicit function to differentiate. Or you could go about it using implicit differentiation!
Then use the quotient rule!
Or
Rearrange and remember to substitute in y again from the fact that .