Chapter 2.4, Problem 37E

To determine

Verify that the data show a quadratic relationship. Predict the number of tiles in the 12th figure.

Answer to Problem 37E

The equation is $y(x)=x^2+x-{1.}^{}$

Number of tiles in the 12th figure $y(12)=155.$ .

Explanation of Solution

Given information: The table shows the number of tiles in each figure.

  

Calculation :

$y(1)$		$y(2)$		$y(3)$		$y(4)$
1		5		11		19
	4		6		8
		2		2		8

Because second differences are constant so we can modal the data with a quadratic function .

  $y(x)=a{x}^2+bx+c.$.......(1)

  $\begin{array}{l}$ y(1)=a\times {(1)}^2+b\times 1+c.$1=a+b+c.\\ a+b+c=\mathrm{1........}(2)\\ $ y(2)=a\times {(2)}^2+b\times 2+c.$5=4a+2b+c.\\ 4a+2b+c=\mathrm{5......}(3)\\ $ y(3)=a\times {(3)}^2+b\times 3+c.$11=9a+3b+c.\\ 9a+3b+c=\mathrm{11......}(4)\\ \text{solving}\text{(2)}\text{and}\text{(3)}and(4).\\ a=1,b=1,c=1.\\ \text{Putting}a=1,b=1,c=-1\text{in}\text{equation}(1)\\ y(x)=x^2+x-{1.}^{}\end{array}$

The equation is $y(x)=x^2+x-{1.}^{}$

  $\begin{array}{l}y(12)={12}^2+12-1.\\ y(12)=144+12-1.\\ y(12)=155.\end{array}$