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

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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 2.9 4.1 5.6 6.3 7 8 10 13.9 Number of Drops 42 21 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

. Press 1. Move the cursor to On and select it by pressing ENTER. Move the cursor to the line labeled Type. The first icon represents a scatter plot. Select the scatter plot type and press ENTER. Move the cursor to the line labeled Xlist and press 2nd, then LIST and select L1 to enter it as the variable x. Move the cursor to the line labeled Ylist and press 2nd, then LIST and select L2 to enter it as the variable y. Move the cursor to the line labeled Mark and press ENTER to select the first icon. This is the mark that will be used for each point in the scatter plot.Hit GRAPH to display the scatter plot.Return to and clear the home screen by pressing 2nd, QUIT, and then CLEAR. Hit the STAT button, move the cursor to CALC, and press ENTER. Move the cursor down to find PwrReg and select it by pressing ENTER. Hit ENTER again to receive the values for

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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