Question

(a) A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacturer a package of screw A, while it takes 6 minutes on automatic and 3 minutes on the hand operatedmachines to manufacture a package of screw B. Each machine is available for at the most 4 hours (240minutes) on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs.10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

$\left(b\right)Provethat\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=abc+bc+ca+ab$

Answer 1

Let x, y be number of packs of

(a) Screw A and Screw B respectively

$\Rightarrow z=7x+10y$ for graph

$x\ge 0,y\ge 0\text{and}4x+6y\le 240$ for corner point .

$6x+3y\le 240$

Using corner point method the maximum profit occurs at x = 30, y = 20

$\Rightarrow$ maximum profit = Rs 410

$\therefore {\int }_0^{2a}f\left(x\right)dx={\int }_0^{a}f\left(x\right)dx+{\int }_0^{a}f(2a-x)dx=\{\begin{array}{cc}2{\int }_0^{a}f\left(x\right)dx,& iff(2a-x)=f\left(x\right)\\ 0,& iff(2a-x)=-f\left(x\right)\end{array}I={\int }_0^{a}f\left(x\right)dxf\left(x\right)=co{s}^{5}x\Rightarrow f(2\pi -x)=co{s}^5(2\pi -x)=co{s}^{5}xI=2{\int }_0^{a}co{s}^{5}xdxf(\pi -x)=co{s}^5(\pi -x)=-co{s}^{5}x=-f\left(x\right)\Rightarrow I=0$

$\left(b\right)\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=abc\left|\begin{array}{ccc}1+\frac{1}{a}& \frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}& 1+\frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|C_1\to C_1+C_2+C_3=abc\left|\begin{array}{ccc}1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}& \frac{1}{b}& \frac{1}{c}\\ 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}& 1+\frac{1}{b}& \frac{1}{c}\\ 1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|$

$=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\left|\begin{array}{ccc}1& \frac{1}{b}& \frac{1}{c}\\ 1& 1+\frac{1}{b}& \frac{1}{c}\\ 1& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|R_2-R_1,R_3-R_1=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\left|\begin{array}{ccc}1& \frac{1}{b}& \frac{1}{c}\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|=abc(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1)=abc+ab+bc+ca$