Calculator Technique for Solving Volume Flow Rate Problems in Calculus

The following models of CASIO calculator may work with this method: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus.
 

The following calculator keys will be used for the solution

Name Key Operation
Shift shift.jpg SHIFT
Mode mode-setup.jpg MODE
Name Key Operation
Stat 1-stat.jpg SHIFT → 1[STAT]
AC ac-off.jpg AC

 

This is one of the series of post in calculator techniques in solving problems. You may also be interested in my previous posts: Calculator technique for progression problems and Calculator technique for clock problems; both in Algebra.
 

Flow Rate Problem
Water is poured into a conical tank at the rate of 2.15 cubic meters per minute. The tank is 8 meters in diameter across the top and 10 meters high. How fast the water level rising when the water stands 3.5 meters deep.
 

Traditional Solution
flow-rate-cone.gifrh=410

r=25h
 

Volume of water inside the tank
V=13πr2h

V=13π(25h)2h

V=475πh3
 

Differentiate both sides with respect to time
dVdt=425πh2dhdt

2.15=425πh2dhdt
 

When h = 3.5 m
2.15=425π(3.52)dhdt

dhdt=0.3492m/min           answer
 

Solution by Calculator

 

flow-rate-cone-caltech.gifMODE &rarr; 3:STAT &rarr; 3:_+cX<sup>2</sup>
 

X Y
0 0
10 π42
5 π22

AC &rarr; 2.15 &divide; 3.5y-caret = 0.3492           answer
 

To input the 3.5y-caret above, do
3.5 &rarr; SHIFT &rarr; 1[STAT] &rarr; 7:Reg &rarr; 6:y-caret
 

What we just did was actually v = Q / A which is the equivalent of dhdt=dV/dtA for this problem.
 

Problem
Water is being poured into a hemispherical bowl of radius 6 inches at the rate of x cubic inches per second. Find x if the water level is rising at 0.1273 inch per second when it is 2 inches deep?
 

Traditional Solution
flow-rate-hemisphere.gifVolume of water inside the bowl
V=13πh2(3rh)

V=13πh2[3(6)h]

V=13π(18h2h3)
 

Differentiate both sides with respect to time
dVdt=13π(36h3h2)dhdt
 

When h = 2 inches, dh/dt = 0.1273 inch/sec
dVdt=13π[36(2)3(22)](0.1273)

x=7.9985in3/sec           answer
 

Calculator Technique
flow-rate-hemisphere-caltech.gifMODE &rarr; 3:STAT &rarr; 3:_+cX<sup>2</sup>
 

X Y
0 0
6 π62
12 0

AC &rarr; 0.1273 &times; 2y-caret = 7.9985           answer
 

I hope you enjoy this post. Next time you solve problems involving flow rate, try to use this calculator technique to save time.