Deck 13: Fibonacci Numbers and the Golden Ratio: Tales of Rabbits and Gnomons

Question 1

A painter is constructing a rectangular canvas for her next masterpiece. She plans to construct the canvas so that the ratio of the larger side to the smaller side is equivalent to the golden ratio. If the smaller side is 233 cm in length, then what is the length of the larger side in centimeters? Round the
answer to the nearest whole number.

Answer 1

377 cm

Question 2

Given that F₂₀=6765 and F₂₂=17711 , find F₁₉ .

Answer 2

4181

Question 3

The two polygons P and P' shown below are similar. If the area of P is 25, then find the area of P'.

Answer 3

56.25

Question 4

Use Binet's formula is the golden ratio, to find the thirty-second Fibonacci number F₃₂ .

Question 5

The two polygons P and P' shown below are similar. If the perimeter of P is 23, then find the perimeter of P'.

Question 6

A painter is constructing a rectangular canvas for her next masterpiece. She plans to construct the canvas so that the ratio of the larger side to the smaller side is equivalent to the golden ratio. If the larger side is 267 cm in length, then what is the length of the smaller side in centimeters? Round the answer to the nearest whole number.

Question 7

Suppose that you are told that , find an approximate value for https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Write your answer in scientific notation with six significant digits. (Hint: Use the fact that values of )

Question 8

Find the product of the first six numbers in the Fibonacci sequence.

Question 9

If represents the N^th Fibonacci number, then the Fibonacci sequence can be constructed by the recursive rule https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Is the following rule always correct? Explain.

Question 10

The golden ratio is an irrational number. Explain what it means for a number to be irrational.

Question 11

If represents the Fibonacci number, then the Fibonacci sequence can be constructed by the recursive rule
. Is the following rule correct? Explain.

Question 12

Rectangle A has dimensions of 2 by 8. Rectangle B is a gnomon to rectangle
A. What are the dimensions of rectangle B?

Question 13

Use Binet's formula ,where is the golden ratio, to find the forty third Fibonacci number .

Question 14

Find the value x so that the shaded figure is a gnomon to the white triangle.

Question 15

Suppose that and are two similar rectangles where the width of is a and the width of is 4a . If the area of is 64 square inches, then what is the area of in square inches?

Question 16

Rectangle A has dimensions of 3 by 6. Rectangle B is a gnomon to rectangle A. What are the
dimensions of rectangle B?

Question 17

Find the sum of the first six numbers in the Fibonacci sequence.

Question 18

Knowing that FN represents the Nth Fibonacci number, and given that F20 = 6765 and F21 = 10946 , find F24 .

Question 19

Find the value of y so that the shaded figure is a gnomon to the white triangle.

Question 20

Suppose that R and R' are two similar rectangles where the width of R is a and the width of R' is 6Pa . If the perimeter of R is 42 inches, then what is the perimeter of R' in inches?

Question 21

Find the following sum.

A) 25
B) 33
C) 65
D) 95
E) None of the above.

Question 22

Find the value x so that the shaded figure is a gnomon to the white triangle.

A) 24
B) 36
C) 48
D) 72
E) None of the above.

Question 23

Find the value y so that the shaded figure is a gnomon to the white triangle.

(a) 24
(b) 26
(c) 36
(d) 42
(e) None of the above.

Question 24

A painter is constructing a rectangular canvas for her next masterpiece. She plans to construct the canvas so that the ratio of the larger side to the smaller side is equivalent to the golden ratio. If the smaller side is 416 \mathrm{~cm} in length, then what is the length of the larger side in centimeters? Round the answer to the nearest whole number.

A) 257 cm
B) 416 cm
C) 673 cm
D) 852 cm
E) None of the above.

Question 25

Given that =4181 and =10946 , find .

A) 6765
B) 17711
C) 21892
D) 46368
E) None of the above.

Question 26

Suppose that and are two similar rectangles where the width of is 3 a and the width of is 5 a . If the perimeter of is 42 inches, then what is the perimeter of in inches?

A) 21
B) 48
C) 56
D) 70
E) None of the above.

Question 27

Given that =6765 and =10946 , find .

A) 2584
B) 4181
C) 8856
D) 17711
E) None of the above.

Question 28

Suppose that you are told that , find an approximate value for . Write your answer in scientific notation with six significant digits. (Hint: Use the fact that values of N .)

A) 5.28522 X 10337
B) 1.00347 X 10338
C) 3.83292 X 10337
D) 2.16352 X 10337
E) None of the above.

Question 29

A painter is constructing a rectangular canvas for her next masterpiece. She plans to construct the canvas so that the ratio of the larger side to the smaller side is equivalent to the golden ratio. If the larger side is 416 cm in length, then what is the length of the smaller side in centimeters? Round the answer to the nearest whole number.

A) 257 cm
B) 416 cm
C) 673 cm
D) 852 cm
E) None of the above.

Question 30

If represents the Fibonacci number, then the Fibonacci sequence can be constructed by the recursive rule
- Which of the following equations is valid?

A)
B)
C)
D)
E) None of the above.

Question 31

Find the following product.

A) 254
B) 322
C) 504
D) 953
E) None of the above.

Question 32

Rectangle A has dimensions of 6 by 18. Rectangle B is a gnomon to rectangle A. What are the dimensions of rectangle B?

A) 18 by 45
B) 18 by 48
C) 18 by 52
D) 18 by 54
E) None of the above.

Question 33

Suppose that and are two similar rectangles where the width of is 3a and the width of is 7a . If the perimeter of is 18 inches, then what is the perimeter of in inches?

A) 6
B) 18
C) 36
D) 42
E) None of the above.

Question 34

If the area of is 18 and the area of is 72 , then what would be the value of in order for the two polygons and to be similar?

A) 8
B) 12
C) 16
D) 24
E) None of the above.

Question 35

If represents the Fibonacci number, then the Fibonacci sequence can be constructed by the recursive rule ,
. Which of the following equations is not valid?

A)
B)
C)
D)
E) None of the

Question 36

Suppose that you are told that
, find an approximate value for https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Write your answer in scientific notation with six significant digits. (Hint: Use the fact that values of N . )

A) 5.28522 X 10³³⁷
B) 1.00347 X 10³³⁸
C) 3.83292 X 10³³⁷
D) 2.16352 X 10³³⁷
E) None of the above.

Question 37

Use Binet's formula , where is the golden ratio, to find the forty-third Fibonacci number https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/.

A) 125214525
B) 325858451
C) 412569854
D) 433494437
E) None of the above.

Question 38

If the perimeter of is 24 and the perimeter of is 72 , then what would be the value of or in order for the two polygons and to be similar?

A) 8
B) 12
C) 16
D) 24
E) None of the above.

Question 39

Use Binet's formula , where is the golden ratio, to find the thirty-fifth Fibonacci number F₃₅ .

A) 1252522
B) 3528965
C) 7542583
D) 9227465
E) None of the above.

Question 40

Rectangle A has dimensions of 4 by 16. Rectangle B is a gnomon to rectangle A. What are the dimensions of rectangle B?

A) 16 by 45
B) 16 by 60
C) 16 by 64
D) 16 by 72
E) None of the above.