Water is being poured into a hemispherical bowl of radius 3 inch at the rate of 1 inch^3/s. How fast is the water level rising when the water is 1 inch deep ?

1 Answer
Mar 23, 2018

#1/(5pi} "in/s" ~~ 0.064 " in/s"#

Explanation:

When the water level is one inch deep, its upper surface is a circle whose radius #r# (in inches) is given by

#r^2= 3^2-(3-1)^2 = 9-4=5#

Thus the cross sectional area of the surface is #pi r^2 = 5pi " in"^2#

In a small time #delta t# seconds, the volume increases by #delta t " in"^3#. You can think of this extra volume as that of a cylinder of height #delta h# inches and cross sectional area #5pi " in"^2#. (True - the extra volume is actually a slice cut off from a sphere - and not a cylinder, but the difference disappears as #delta t to 0# )

Thus

#delta h = {delta t}/{5pi}#

Thus, the rate at which height is increasing is #1/(5pi} "in/s"#