Chapter 4, Problem 12CT

To determine

To determine: A polynomial model for the data using the finite difference. Also, predict the profit after 7 months using the model.

Answer to Problem 12CT

The required model is $f\left(x\right)=6x^2-23.6x+22.8$ and the profit after 7 months is 151.6 dollars.

Explanation of Solution

Given information:

The table for profits p of the business in the first 5 months:

Month, t	1	2	3	4	5
Profit, p	4	2	6	22	56

The given table is:

Month, t	1	2	3	4	5
Profit, p	4	2	6	22	56

Write the table by using finite difference:

$f\left(1\right)$		$f\left(2\right)$		$f\left(3\right)$		$f\left(4\right)$		$f\left(5\right)$
4		2		6		22		56
	−2		4		16		34
		6		12		18
			6		6

From the above table write the required model as:

  $f\left(x\right)=6x^2-23.6x+22.8$

Substitute 7 for x in the above model.

  $\begin{array}{c}f\left(7\right)=6{\left(7\right)}^2-23.6\left(7\right)+22.8\\ f\left(7\right)=6\left(49\right)-165.2+22.8\\ f\left(7\right)=294-165.2+22.8\\ f\left(7\right)=151.6\end{array}$

Hence, the required model is $f\left(x\right)=6x^2-23.6x+22.8$ and the profit after 7 months is 151.6 dollars.