Chapter 2, Problem 11CT

To determine

To calculate: To find a quadratic function that best models the data. Predict the proce of gold in the year 2016.

Answer to Problem 11CT

The expected price of gold in 2016 is 5332.16

Explanation of Solution

Given information: Following data is given,

x	y
0	603.46
1	695.39
2	871.96
3	972.35
4	1224.53
5	1571.52

Here, 0 denotes the price of gold in 2006.

Calculation:

The data can be modeled by the equation,

  $y=a{x}^2+bx+c$

Since three unknowns are to be determined, substitute three values from the model data and simplify for a, b and c

  $\begin{array}{c}y=a{x}^2+bx+c\\ 603.46=a{\left(0\right)}^2+b\left(0\right)+c\left(x=0,y=603.46\right)\\ c=603.46\mathrm{......}\left(1\right)\\ 695.39=a{\left(1\right)}^2+b\left(1\right)+c\left(x=1,y=695.39\right)\\ a+b+c=695.39\\ a+b+603.46=695.39\\ a+b=\text{695}\text{.39-603}\text{.46}\\ \text{a}+b=91.92\mathrm{......}\left(2\right)\\ 871.96=a{\left(2\right)}^2+b\left(2\right)+c\left(x=2,y=871.96\right)\\ 4a+2b+c=871.96\\ 4a+2b=871.96-603.46\\ 4a+2b=268.5\\ \text{2a}+b=134.25\mathrm{......}\left(3\right)\end{array}$

Simplify equations (2) and (3),

  $\begin{array}{l}a+b=91.92\\ 2a+b=134.25\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\\ ---\\ -a=91.92-134.25\\ -a=-42.33\\ a=42.33\end{array}$

Substitute the value of a in equation (2),

  $\begin{array}{c}a+b=91.92\\ 42.33+b=91.92\\ b=91.92-42.33\\ =49.59\end{array}$

Substitute the values of a, b, and c in $y=a{x}^2+bx+c$ ,

  $\begin{array}{l}y=a{x}^2+bx+c\\ y=42.33x^2+49.59x+603.46\end{array}$

Now, 2016 is 10 years after 2006, therefore, the price of gold in 2016 can be obtained by substituting $x=10$ in the obtained model,

  $\begin{array}{c}y=42.33x^2+49.59x+603.46\\ y\left(2016\right)=42.33{\left(10\right)}^2+49.59\left(10\right)+603.46\\ =4233+495.9+603.46\\ =5332.36\end{array}$