Question

How much energy (in million $\left)andmanufacturing\right(inmillion$) must be produced to have $\$8$ million worth of energy and $\$5$ million worth of manufacturing available for consumer use?

• A. Energy $11$,
  Manufacturing $5.75$
• B. Energy $17.7$,
  Manufacturing $10.5$
• C. Energy $21$,
  Manufacturing $10.5$
• D. Energy $8$,
  Manufacturing $5.75$

Answer 1

The correct option is B Energy $17.7$,

Manufacturing $10.5$
Let the amount of energy be $x$ and manufacturing be $y$

Given that energy available for consumer use is $8$ million worth

Given that manufacturing available for consumer use is $5$ million worth

The energy which is consumed is $0.4x+0.25y$

The manufacturing which is consumed is $0.25x+0.1y$

The left over energy and manufacturing available for consumer use can be calculated by using below mentioned equation

$\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]-\left[\begin{array}{cc}0.4& 0.25\\ 0.25& 0.1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 5\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}0.6& -0.25\\ -0.25& 0.9\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 5\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}0.6& -0.25\\ -0.25& 0.9\end{array}\right]}^{-1}\left[\begin{array}{c}8\\ 5\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{0.4775}\left[\begin{array}{cc}0.9& 0.25\\ 0.25& 0.6\end{array}\right]\left[\begin{array}{c}8\\ 5\end{array}\right]=\frac{1}{0.4775}\left[\begin{array}{c}8.45\\ 5\end{array}\right]=\left[\begin{array}{c}17.7\\ 10.5\end{array}\right]$

Therefore energy available is $17.7$ and manufacturing available is $10.5$

Hence, the correct option is $B$.