A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost - Maths - Application of Derivatives

Question

A water tank has the shape of an inverted right circular cone with
its axis vertical and vertex lowermost Its semi-vertical angle is
tan inverse(0 5) Water is poured into it at a constant rate of 5
cubic metre per hour Find the rate at which the level of water is
rising at the instant when the depth of water in the tank is 4 m

Udbhav Singh · Accepted Answer

Dear Student,We know, tanα = rh 
1Here, r - radiush- hieghtα-semi vertical angle Also, α=tan-10
5Thus, Tan(tan-10 5)=rh  Using 1⇒0
5=rhThus, r=h2Also, Volume of Cone, V = 13πr2h= 13πh22(h)= 112πh3Now, dVdt=dVdh×dhdt
Using Chain RuledVdt=d112πh3dr×dhdt=14πh2dhdtNow, It is given that Water is poured into it at a constant rate of 5 cubic metre per hourThus, dVdt=5 m3/hrThus, 14πh2dhdt=5⇒dhdt=20πh2Thus, dhdt at h=4m⇒dhdt=20π(4)2=54π=0
39 m/hThus, At the instant when h=4m , Water level is rising at a rate of 0
39m per hour

Regards