Calculator Technique for Solving Volume Flow Rate Problems in Calculus
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Transcript of Calculator Technique for Solving Volume Flow Rate Problems in Calculus
![Page 1: Calculator Technique for Solving Volume Flow Rate Problems in Calculus](https://reader031.fdocuments.in/reader031/viewer/2022020717/577cce181a28ab9e788d4c78/html5/thumbnails/1.jpg)
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
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
Volume of water inside the tank
Differentiate both sides with respect to time
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When h = 3.5 m
answer
Solution by Calculator
HideClick here to show or hide the concept behind this technique
Understand why this calculator technique works In Hydraulics, discharge or volume flow rate is given by the formula
Where
= the discharge or volume flow rate
= velocity of flow
= cross-sectional area of flow
The equivalent of the above elements in Calculus are:
where V is the volume and dV/dt is the volume flow (time) rates and
where h, x, y, s are distances and v is velocity.
Thus, the formula Q = vA can be written as
which is the formula we are going to use in our calculator.
For the area A The general prismatoid is a solid in which the area of any section, say Ay, parallel to and at a
distance y from a fixed plane can be expressed as a polynomial in y not higher than third degree,
or
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Common solids like cone, prism, cylinder, frustums, pyramid, and sphere are actually
prismatoids in which any area parallel to a base is at most a quadratic function in height y. Thus,
We can therefore use the Quadratic Regression in STAT mode of the calculator to find the area
in relation with its distance from a predefined plane.
MODE → 3:STAT → 3:_+cX2
X Y
0 0
10 π42
5 π22
AC → 2.15 ÷ 3.5y-caret = 0.3492 answer
To input the 3.5y-caret above, do
3.5 → SHIFT → 1[STAT] → 7:Reg → 6:y-caret
What we just did was actually v = Q / A which is the equivalent of 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
Differentiate both sides with respect to time
When h = 2 inches, dh/dt = 0.1273 inch/sec
answer
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Calculator Technique
MODE → 3:STAT → 3:_+cX2
X Y
0 0
6 π62
12 0
AC → 0.1273 × 2y-caret = 7.9985 answer
- See more at: http://www.mathalino.com/blog/romel-verterra/calculator-technique-solving-volume-
flow-rate-problems-calculus#sthash.55HsW1Xj.dpuf