25. A manufacturer makes two types of toys, A and B. Three machines are needed for this purpose, and the time (in minutes) required for each toy on the machines is given below:

Types of Toys	Machines
	I	II	III
A	12	18	6
B	6	0	9

Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is ?. 7.50 and that on each toy of type B is ?. 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

Let x and y toys of type A and type B be manufactured in a day, respectively.

The given problem can be formulated as given below:

$Maximisez=7.5x+5y\dots \dots \dots \dots ..\left(i\right)$

Subject to the constraints,

$\begin{array}{}\end{array}2x+y?60\dots \dots \dots \dots .\left(ii\right)$

$x?20\dots \dots \dots ..\left(iii\right)$

$2x+3y?120\dots \dots \dots ..\left(iv\right)$

$x,y?0\dots \dots \dots \dots \dots .\left(v\right)$

The feasible region determined by the constraints is given below:

A (20, 0), B (20, 20), C (15, 30) and D (0, 40) are the corner points of the feasible region.

The values of z at these corner points are given below:

Corner Point	z = 7.5x + 5y
A (20, 0)	150
B (20, 20)	250
C (15, 30)	262.5	Maximum
D (0, 40)	200

262.5 at (15, 30) is the maximum value of z.

Hence, the manufacturer should manufacture 15 toys of type A and 30 toys of type B to maximise the profit.

<p>Let x and y toys of type A and type B be manufactured in a day, respectively.</p><p>The given problem can be formulated as given below:</p><p><span title="Click to copy mathml"><math><mrow><mi>M</mi><mi>a</mi><mi>x</mi><mi>i</mi><mi>m</mi><mi>i</mi><mi>s</mi><mi>e</mi><mi>z</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p><p>Subject to the constraints, </p><p><span title="Click to copy mathml"><math><mrow><mtable columnalign="left"></mtable><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>? </mo><mn>6</mn><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi>x</mi><mo>? </mo><mn>2</mn><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>i</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>? </mo><mn>1</mn><mn>2</mn><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>i</mi><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mrow><mi>x</mi><mo>, </mo><mi>y</mi><mo>? </mo><mn>0</mn><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mrow><mo> (</mo><mrow><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p>The feasible region determined by the constraints is given below:</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1732774107phpyeDZVv_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1732774107phpyeDZVv.jpeg" alt="" width="291" height="230"></picture></div></div><p>A (20, 0), B (20, 20), C (15, 30) and D (0, 40) are the corner points of the feasible region.</p><p>The values of z at these corner points are given below:</p><table><tbody><tr><td><p>Corner Point</p></td><td><p>z = 7.5x + 5y</p></td><td>&nbsp;</td></tr><tr><td><p>A (20, 0)</p></td><td><p>150</p></td><td>&nbsp;</td></tr><tr><td><p>B (20, 20)</p></td><td><p>250</p></td><td>&nbsp;</td></tr><tr><td><p>C (15, 30)</p></td><td><p>262.5</p></td><td><p>Maximum</p></td></tr><tr><td><p>D (0, 40)</p></td><td><p>200</p></td><td>&nbsp;</td></tr></tbody></table><p>262.5 at (15, 30) is the maximum value of z.</p><p>Hence, the manufacturer should manufacture 15 toys of type A and 30 toys of type B to maximise the profit.</p>

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