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Session 9: Solutions
 
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Solutions for Session 9, Part C

See solutions for Problems: C1 | C2 | C3 | C4


Problem C1

On most pineapples, all three numbers will be consecutive Fibonacci numbers: 8, 13, and 21.

<< back to Problem C1


 

Problem C2

The ratios seem to be approaching one number, which is about 1.618, to three decimal places.

<< back to Problem C2


 

Problem C3

If the pattern continues, this ratio should be fairly close to the ratios found in the table in Problem C2; it should also be very close to the ratio of the other consecutive Fibonacci numbers around it.

<< back to Problem C3


 

Problem C4

Consider the ratio of sides in each golden rectangle. In Rectangle 1, the ratio is ø to 1. In Rectangle 1 + 2, the ratio is ø + 1 to ø. These are similar rectangles, so the ratios must be equal:

Cross-multiplying and simplifying the equation gives us a quadratic equation: ø2 - ø - 1 = 0. This equation does not factor, so we must use the quadratic formula to find the value of ø; the two possible values are

.
Since the side of a rectangle can't be negative,

.

Evaluating this on a calculator gives us the decimal 1.618 to three decimal places, as expected.

<< back to Problem C4


 

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