Question

(b) एक फैक्ट्री में टेनिस रैकट और क्रिकेट बैट बनाये जाते हैं। एक टेनिस रैकेट को बनाने के लिए मशीन
पर 1 घंटा और हाथ द्वारा कार्य के लिए 13 घंटों का समय लगता है जबकि एक क्रिकेट बैट को बनाने
के लिए मशीन पर 3 घंटे तथा हाथ द्वारा कार्य के लिए घंटे का समय लगता है, उस फैक्ट्री में एक
दिन में मशीन पर अधिक से अधिक 42 घंटे और हाथ द्वारा कार्य के लिए अधिक से अधिक 24 घंट
का समय उपलब्ध है। इन सूचनाओं को रैखिक असमिकाओं द्वारा लिखें तथा इस असमिका निकाय को
आरेखीय विधि से हल करें।​

Answer 1

Step-by-step explanation:

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12th

Maths

Linear Programming

Graphical Solution of a Linear Programming Problems

A factory makes tennis rack...

MATHS

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmans time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time.

(i) What number of rackets and bats must be made if the factory is to work at full capacity?

(ii) If the profit on a racket and on a bat is Rs.20 and Rs.10 respectively, find the maximum profit of the factory when it works at full capacity.

answer:

Let number of tennis racket be made is X and number of cricket bat be made is Y

Since, tennis bat requires 1.5 hours and cricket bat requires 3 hours of machine time. Also, there is maximum 42 hours of machine time available.

∴1.5X+3Y≤42

⇒X+2Y≤28 ...(1)

Since, tennis bat requires 3 hours and cricket bat requires 1 hours of craftmans time. Also, there is maximum 24 hours of craftmans time available.

∴3X+Y≤24 ...(2)

Since, count of an object can't be negative.

∴X≥0,Y≥0 ...(3)

We have to maximize profit of the factory.

Here, profit on tennis racket is 20 Rs and on cricket bat is 10 Rs

So, objective function is Z=20X+10Y

Plotting all the constraints given by equation (1), (2) and (3), we got the feasible region as shown in the image.

Corner points Value of Z=20X+10Y

A (0,14) 140

B (4,12) 200 (maximum)

C (8,0) 160

Hence,

(i) 4 tennis rackets and 12 cricket bats must be made so that factory will work at full capacity.

(ii) Maximum profit of factory will be 200 Rs

solution.