# How many drops does it take for a crow to crack a nut?

Extracts from this document...

Introduction

April 22, 2008

A murder of crows gathers different sizes of nuts. Crows love to eat nuts, but their beaks are not strong enough to crack open some of the shells of these nuts. In order to crack open the shells, they repeatedly drop the nut on a hard surface until it finally opens.

The following table shows the average number of drops it takes to break open a large nut from varying heights.

Height of Drop (m) | 1.7 | 2.0 | 2.9 | 4.1 | 5.6 | 6.3 | 7.0 | 8.0 | 10.0 | 13.9 |

Number of Drops | 42.0 | 21.0 | 10.3 | 6.8 | 5.1 | 4.8 | 4.4 | 4.1 | 3.7 | 3.2 |

Scatter Plot of the Effect of Dropping Large Nuts at Varying Heights

on the Number of Drops It Takes to Crack It Open

where x = Height of Drop (m); x > 0 and y = Number of Drops

The graph models the behavior of a power regression.

Finding the equation algebraically (large nuts)

Power regression formula: f(x) = ax^b where x > 0

Select two points: (2.0, 21.0) and (7.0, 4.4)

For each equation, solve for a: l(21.0) = a(2.0)^b l(4.4) = a(7.0)^b

a = 21.0/2.0^ba = 4.4/7.0^b

Use the transitive property: a = a, therefore 21.0/2.0^b = 4.4/7.0^b

Solve for b: 21.0/2.0^b = 4.4/7.

Middle

Conclusion

on the Number of Drops It Takes to Crack It Open and the Line f(x) = 49.9x^-1.25

where x = Height of Drop (m); x > 0 and y = Number of Drops

The parameters are a and b.

The line f(x) also does not apply to the scatter plot of small nuts. The data points again are above the line f(x). The change needs be made to this line. The value of a must be significantly increased and b may have to be slightly increased.

Finding the equation using a graphing calculator (medium nuts)

For steps, see “Finding the equation using a graphing calculator (large nuts).” Apply data small nuts in place of the large nuts.

Equation: s(x) = 137x^-1.10

Scatter Plot of the Effect of Dropping Small Nuts at Varying Heights

on the Number of Drops It Takes to Crack It Open and the Line s(x) = 137x^-1.10

where x = Height of Drop (m); x > 0 and y = Number of Drops

The parameters are a and b.

This graph fits much better than f(x).

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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