(a) A factory manufactures two types of screws, A and B. Each type of

1.	(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. (b)Prove that∣∣∣∣1+a1111+b1111+c∣∣∣∣=abc(1+1a+1b+1c)=abc+bc+ca+ab
Answer» (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. $(b) P r o v e t h a t ∣ ∣∣ ∣ \begin{matrix} 1 + a & 1 & 1 1 & 1 + b & 1 1 & 1 & 1 + c \end{matrix} ∣ ∣∣ ∣ = a b c (1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = a b c + b c + c a + a b$

(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. (b)Prove that∣∣∣∣1+a1111+b1111+c∣∣∣∣=abc(1+1a+1b+1c)=abc+bc+ca+ab

Answer»

(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.

$(b) P r o v e t h a t ∣ ∣∣ ∣ \begin{matrix} 1 + a & 1 & 1 1 & 1 + b & 1 1 & 1 & 1 + c \end{matrix} ∣ ∣∣ ∣ = a b c (1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) = a b c + b c + c a + a b$

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