Help needed to solve this problem

[sebastien]

Hi, I am not advance enough to solve this.

We are designing a company Logo. The width of the logo is divided by 7 bars. We would like to have the distance between each bars grow exponentially from right to left and yet fit perfectly within the width of the card

I came up with this

X = width

A=first segment width
B=second segment width B=AxZ
C= third segment width C=BxZ

D=CxZ
E=DxZ
F=ExZ
G=FxZ
......

Z=growth multplier ? (sorry my math is poor)

A+B+C+D+E+F+G = X

 

[stapel]

How is "exponentially" defined? Is the distance between the bars being measured along the top of the card? Are the bars included in the widths, or is it really the white-space in between the bars (and the ninth space, being the solid background at the left) that is being measured?

What is the full and exact text of the exercise? What are the complete instructions? How is calculus expected to be used in solving this exercise?

Please be complete. Thank you.

 

[Ishuda]

Just repeating what you said to make sure I have it correct: First you are saying 7 but showing eight. Next, I assume you are including the width of the bar so that from the right hand edge, along the top. So the first bar is along the right hand edge, it is a distance A to the beginning of the second bar, a distance B=A*Z to the beginning of the third bar, ..., a distance G = A*Z⁶ to the other edge of the flag along the top. So the total distance is
A (1 + Z + Z² + ... + Z⁶) = X


The sum is what is know as a geometric progression and has a sum
(Z⁷ - 1) / (Z - 1)
and you want the solution to
Z⁷ - (X/A) (Z - 1) - 1 = 0.

I don't know a general solution but you could try an iterative (Newton's) solution, i.e do an initial estimate and let each next estimate be defined by
Z_n+1 = Z_n - (Z⁷ - (X/A) (Z - 1) - 1) ) / (7 Z⁶ - (X/A) )

For example, suppose we had a flag 60 inches wide and you want the first width to be about 1 inches. The X/A=60. Guess about 2 inches for Z₁, then we have
Z₂ = 2 - (2⁷ - 60 (2 - 1) - 1) / (7 2⁶ -60) = 2 - 67/388 = 1.740627
Z₃ = 1.718554
Z₄ = 1.717224
Z₅ = 1.717219
Z₆ = 1.717219
and we can stop. The last width would be 26.64 inches so A = 1 is probably not a good choice.

If you wanted the first width to be 2 inches, then X/A = 30 and Z=1.47643 and the last width is 20.7 inches.

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