The following models of CASIO calculator may work with this method: fx570ES, fx570ES Plus, fx115ES, fx115ES Plus, fx991ES, and fx991ES Plus.
The following calculator keys will be used for the solution
Name 
Key 
Operation 

Shift 

SHIFT 
Mode 

MODE 

Name 
Key 
Operation 

Stat 

SHIFT → 1[STAT] 
AC 

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
$\dfrac{r}{h} = \dfrac{4}{10}$
$r = \frac{2}{5}h$
Volume of water inside the tank
$V = \frac{1}{3}\pi r^2 h$
$V = \frac{1}{3}\pi (\frac{2}{5}h)^2 h$
$V = \frac{4}{75}\pi h^3$
Differentiate both sides with respect to time
$\dfrac{dV}{dt} = \frac{4}{25}\pi h^2 \dfrac{dh}{dt}$
$2.15 = \frac{4}{25}\pi h^2 \dfrac{dh}{dt}$
When h = 3.5 m
$2.15 = \frac{4}{25}\pi (3.5^2) \dfrac{dh}{dt}$
$\dfrac{dh}{dt} = 0.3492 \, \text{m/min}$ answer
Solution by Calculator
MODE → 3:STAT → 3:_+cX<sup>2</sup>
X 
Y 

0 
0 
10 
π4^{2} 
5 
π2^{2} 
AC → 2.15 ÷ 3.5ycaret = 0.3492
answer
To input the 3.5ycaret above, do
3.5 → SHIFT → 1[STAT] → 7:Reg → 6:ycaret
What we just did was actually v = Q / A which is the equivalent of $\dfrac{dh}{dt} = \dfrac{dV/dt}{A}$ 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
Volume of water inside the bowl
$V = \frac{1}{3}\pi h^2(3r − h)$
$V = \frac{1}{3}\pi h^2 [ \, 3(6)  h \, ]$
$V = \frac{1}{3}\pi (18h^2  h^3)$
Differentiate both sides with respect to time
$\dfrac{dV}{dt} = \frac{1}{3}\pi (36h  3h^2) \dfrac{dh}{dt}$
When h = 2 inches, dh/dt = 0.1273 inch/sec
$\dfrac{dV}{dt} = \frac{1}{3}\pi [ \, 36(2)  3(2^2) \, ] (0.1273)$
$x = 7.9985 \, \text{in}^3\text{/sec}$ answer
Calculator Technique
MODE → 3:STAT → 3:_+cX<sup>2</sup>
AC → 0.1273 × 2ycaret = 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.
Add new comment