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Bean-CAN-tastic Efficiency: Optimal Volume - Coggle Diagram
Bean-CAN-tastic Efficiency: Optimal Volume
Calcuations
Step 1: Given
Measurements of Can
C = 27cm
h = 11cm
Formulas
S.A. = 2πr^2 + 2πrh
V = πr^2h
C = 2πr
Step 2: Required
Find the
optimized volume
using the
limited materials
.
Step 3: Analysis
The
constraint
(limited materials) is the Surface Area of the can. To avoid any errors the circumference was measured using a string.
r = C/2πr = 27/2π
Using r and h find the surface area (constraint) Find the constraint
Isolate for h using the surface area formula
Put h in the Volume formula and find the derivative of the Volume formula
Make the derivative equal to 0 to get the r for the optimized volume. (The derivative of a function represents the slope at each point of the original function and the max/min of a function has a slope of 0)
Step 4: Solution
V = πr^2 (65.73 - r^2)/r simply
V = π r(65.73 - r^2)
V = π (65.73r - r^3) find V'
V' = π(65.73 - 3r^2)
V' = π(65.73 - 3r^2) = 0 V' = (65.73 - 3r^2) = 0
√ r^2 = √(65.73/3)
r = 4.68 cm
Now isolate for h using the S.A. formula
h = (413 - 2πr^2)/2πr simply dived the top by 2π
(65.73 - r^2)/r = h
S.A. = 2π(27/2π)^2 + 2π(27/2π)11 = 413 cm^2
Step 5: Statement
(65.73 - (4.68)^2)/4.68 = h = 9.365 cm
V = π(4.68)(9.365) = 644.4 ml (optimized volume)
Optimized Volume Percentage = Measured Volume/ Optimized volume = 638.13 mL / 644.4 mL x 100% =
99%
How filled the can is = Given volume/ Optimized volume = 540 mL / 644.4 mL x 100% =
83.8%
Graph
Photo Error
https://docs.google.com/document/d/1ko6p-2EnHYlozyYstL2riDDzZ265-ot8MaPFH7unymg/edit?usp=sharing
Possible Errors
Data Variability: Different cans from the same product line may have slight variations in dimensions.
Rounding Discrepancies: Variations in results due to rounding.
Measurement Errors: Inaccurate measurements of the can's dimensions
Grooves: Grooves on the surface of the can, could affect measurements and calculations.
