Chapter 2.4, Problem 24E

To determine

Write an equation for the path of the baseball. Find the height of the baseball after 5 seconds.

Answer to Problem 24E

Equation of the path of the baseball is $f(x)=-2x^2+12x+6.$

  $f(5)=16.$

Explanation of Solution

Given information: A baseball is thrown up in the air. The table shows the heights y

(in feet) of the baseball after x seconds

  

Calculation

Because second differences are constant so we can modal the data with a quadratic function $f(x)=a{x}^2+bx+c.$

$f(0)$		$f(1)$		$f(2)$		$f(3)$
6		22		22		6
	-16		0		16
		-16		-16

  $\begin{array}{l}f(x)=a{x}^2+bx+c\mathrm{.......}(1).\\ f(0)=a\times 0^2+b\times 0+c.\\ 6=0+c.\\ c=6.\\ f(2)=a\times 2^2+b\times 2+6=4a+2b.\\ 4a+2b=16\mathrm{.........}(2)\\ f(4)=a\times 4^2+b\times 4=16a+4b.\\ 16a+4b=\mathrm{16........}(3)\\ \text{solving}\text{equation}\text{(2)}\text{and}\text{(3)}.\\ a=-2\text{and}b=12.\\ \text{putting}a=-2,b=12,c=6\text{in}\text{equation}(1).\\ f(x)=-2x^2+12x+6.\end{array}$

So,

  $f(5)=-2\times 5^2+12\times 5+6=-50+60+6=16.$

So,

Equation of the path of the baseball is $f(x)=-2x^2+12x+6.$

  $f(5)=16.$