Question

Using this table of numbers, develop a formula to find out how many swings it would take to reach level 50.

Answer 1

Answer:

                $\large\boxed{\large\boxed{62,500}}$

Explanation:

By subtracting two consecutive differences you will find a pattern.

The differences in the number of swings, as the level increase, increase by a constant number, equal to 50.

Here are some samples of the differences in the number of swings, to verify that:

• 125 - 75 = 50
• 175 - 125 = 50
• 225 - 175 = 50
• 575 - 5250 = 50

You can prove it with any two consecutive levels from the table and you will always find 50.

If the difference between the number of swings were constant, the relationship between the number of swings and the level would be linear.

The difference of the differences is named second differences. When second differences are constant, the relationship between the variables (level and number of swings) is quadratic.

Then there is a quadratic equation that relates the level and the number of swings.

Such equation, then, is of the form $ax^2+bx+c$

And you can find the coefficients, $a,b,\text{ and }c$ using any trhee points from the table.

The first three points are (1, 25), (2, 100), and (3, 225)

That leads to:

        $25=a+b+c\\ \\ 100=4a+2b+c\\ \\ 225=9a+3b+c$

• To solve it, subtract the first from the second and from the third to obtain two new equations without c:

         $75=3a+b\\ \\ 200=8a+2b$

• Multiply the first by 2, to eliminate b:

            $150=6a+2b\\ \\ 200=8a+2b$

• Subtract the first from the second:

           $50=2a$

           $a=25$

• Solve for b from any of the equations:

              $75=3a+b\\ \\ b=75-3a\\ \\ b=75-3(25)=75-75-0$

• Now you can solve for c from the first equation:

             $25=a+b+c\\ \\ c=25-a-b\\ \\ c=25-25=0$

And you have your equation:

           $swings=25x^2$

For level 50, replace x with 50 to find the number of swings it would take to reach level 50:

             $swings=25x^2=25(50)^2=62,500$