Chapter 9, Problem 22CLT

To determine

To find: the amount worth of which the gym shoes, jogging shoes, and walking shoes were sold that year.

Answer to Problem 22CLT

Gym Shoes: $2539 million

Jogging shoes: $2362 million

Walking shoes: $4418 million

Explanation of Solution

The matrix given below shows the percents (in decimal form) of the total amounts spent on three types of footwear in a recent year.

  $\begin{array}{l}\begin{array}{ccc}\begin{array}{l}\text{ Gym }\\ \text{ shoes }\end{array}& \begin{array}{l}\text{Jogging}\\ \text{shoes}\end{array}& \begin{array}{l}\text{Walking}\\ \text{shoes}\end{array}\end{array}\\ A=\text{Age group}\{\begin{array}{c}14-17\\ 18-24\\ 25-34\end{array}\left[\begin{array}{ccc}0.079& 0.064& 0.029\\ 0.050& 0.060& 0.022\\ 0.103& 0.259& 0.085\end{array}\right]\end{array}$

The total amounts (in millions of dollars) spent by the age groups on the three types of footwear were $479.88 (14−17 age group), $365.88 (18−24 age group), and $1248.89 (25−34 age group). This can be represented in matrix form as:

  $B=\text{Age group}\{\begin{array}{c}14-17\\ 18-24\\ 25-34\end{array}\left[\begin{array}{c}479.88\\ 365.88\\ 1248.89\end{array}\right]$

Let the total amount spent on gym shoes is $ x , amount spent on jogging shoes is $ y , amount spent on walking shoes is $ z .

Now, convert the total amount spent on three type of shoes in matrix form as:

  $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Thus, the given problem can be expressed in matrix form as:

  $\begin{array}{l}AX=B\\ \left[\begin{array}{ccc}0.079& 0.064& 0.029\\ 0.050& 0.060& 0.022\\ 0.103& 0.259& 0.085\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}479.88\\ 365.88\\ 1248.89\end{array}\right]\end{array}$

Now use Cramer’s rule to solve this problem.

Determinant of the coefficient matrix A is:

  $\begin{array}{c}D=\left|\begin{array}{ccc}0.079& 0.064& 0.029\\ 0.050& 0.060& 0.022\\ 0.103& 0.259& 0.085\end{array}\right|\\ =0.079\left|\begin{array}{cc}0.060& 0.022\\ 0.259& 0.085\end{array}\right|-0.064\left|\begin{array}{cc}0.050& 0.022\\ 0.103& 0.085\end{array}\right|+0.029\left|\begin{array}{cc}0.050& 0.060\\ 0.103& 0.259\end{array}\right|\\ =0.000022112\end{array}$

Since $D\ne 0$ , Cramer’s rule can be applied.

Consider,

  $\begin{array}{c}{D}_x=\left|\begin{array}{ccc}479.88& 0.064& 0.029\\ 365.88& 0.060& 0.022\\ 1248.89& 0.259& 0.085\end{array}\right|\\ =.05613776\end{array}$

  $\begin{array}{c}{D}_y=\left|\begin{array}{ccc}0.079& 479.88& 0.029\\ 0.050& 365.88& 0.022\\ 0.103& 1248.89& 0.085\end{array}\right|\\ =0.0522384\end{array}$

  $\begin{array}{c}{D}_z=\left|\begin{array}{ccc}0.079& 0.064& 479.88\\ 0.050& 0.060& 365.88\\ 0.103& 0.259& 1248.89\end{array}\right|\\ =0.09768848\end{array}$

Now determine x ,

  $\begin{array}{c}x=\frac{D_x}{D}\\ =\frac{.05613776}{0.000022112}\\ \approx 2538.79\end{array}$

Now determine y ,

  $\begin{array}{c}y=\frac{D_y}{D}\\ =\frac{0.0522384}{0.000022112}\\ \approx 2362.44\end{array}$

Now determine z ,

  $\begin{array}{c}z=\frac{D_z}{D}\\ =\frac{0.09768848}{0.000022112}\\ =4417.89\end{array}$

So, the solution of the given system is,

  $\left(x,y,z\right)\approx \left(2539,2362,4418\right)$

Thus, amount spent on three type of shoes is:

Gym Shoes: $2539 million

Jogging shoes: $2362 million

Walking shoes: $4418 million