Question

3(a) An amount of Rs. 8,000 is distributed into three investments at the rate of 7%, 8% and 9% per annum respectively. The total annual income is Rs. 317.50 and the annual income from the first investment is Rs. 5 more than the income from the second. Find the amount of each investment using matrix algebra.

3(b) A wholesaler of pencils charges Rs. 24 per dozen on orders of 50 dozens or less. For orders in excess of 50 dozens, the price is reduced by 20 paise per dozen in excess of 50 dozens. Find the size of the order that maximizes his total revenue.

3(c) A computer is purchased for Rs. 50,000. It is estimated to depreciate at 10% per annum every year. Find its scrap value at the end of 12 years. 

Answer 1

a)

Let investments be x, y, z.

$0.6x+0.7y+0.8z=317.5$

$x+y+z=8000$

$0.07x=0.08y+5$

$\begin{bmatrix} 1 & 1 & 1 \\ 7 & 8 & 9 \\ 7 & -8 & 0 \end{bmatrix} \ \begin{bmatrix} 8000 \\ 31750 \\ 500 \end{bmatrix}$

$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 7 & -8 & 0 \end{bmatrix} \ \begin{bmatrix} 8000 \\ -24250 \\ 500 \end{bmatrix}$

$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1& 2 \\ 0 & -15 & -7 \end{bmatrix} \ \begin{bmatrix} 8000 \\ -24250 \\ -55500 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \ \begin{bmatrix} \frac{322500}{23} \\ \frac{280750}{23} \\ \frac{-419250}{23} \end{bmatrix}$

$z<0$  then the problem has no solution.

Suppose that an amount of Rs. 4,000 is distributed into three investments

$0.6x+0.7y+0.8z=317.5$

$x+y+z=8000$

$0.07x=0.08y+5$

$\begin{bmatrix} 1 & 1 & 1 \\ 7 & 8 & 9 \\ 7 & -8 & 0 \end{bmatrix} \ \begin{bmatrix} 4000 \\ 31750 \\ 500 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \ \begin{bmatrix} 1500 \\ 1250 \\ 1250 \end{bmatrix}$

x=1500, y=1250, z=1250.

b) Let $x=$the size of the order

$Revenue= 24x$ if $x<50$

$Revenue= x*(24-0.2*(x-50))$ if $x>50$

For maximum Revenue will find derivative

$x=− 2(−0.2) 34 ​ =85$

Order in $85$ dozen maximizes his total revenue.

с) $50000*(1-0.1)^{12}=14121.48$

Scrap value at the end of 12 years is Rs. $14121.48.$