30. Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Let the fruit grower use x bags of brand P and y bags of brand Q, respectively.

The problem can be formulated as given below:

$Maximisez=3x+3.5y\dots \dots \dots ..\left(i\right)$

Subject to the constraints,

$\begin{array}{}\end{array}x+2y \ge 240\dots \dots \dots ..\left(ii\right)$

$x+0.5y\ge 90\dots \dots \dots ..\left(iii\right)$

$1.5x+2y\le 310\dots \dots \dots ..\left(iv\right)$

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

The feasible region determined by the system of constraints is given below:

A (140, 50), B (20, 140) and C (40, 100) are the corner points of the feasible region.

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

Corner Point	z = 3x + 3.5y
A (140, 50)	595	Maximum
B (20, 140)	550
C (40, 100)	470

The maximum value of z is 595 at (140, 50).

Hence, 140 bags of brand P and 50 bags of brand Q should be used to maximise the amount of nitrogen.

Thus, the maximum amount of nitrogen added to the garden is 595 kg.

<p>Let the fruit grower use x bags of brand P and y bags of brand Q, respectively.</p><p>The 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>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn><mi>y</mi><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><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>&nbsp;</mo><mo>≥</mo><mn>2</mn><mn>4</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>0</mn><mo>.</mo><mn>5</mn><mi>y</mi><mo>≥</mo><mn>9</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>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>≤</mo><mn>3</mn><mn>1</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><mrow><mo> (</mo><mrow><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></span></p><p>The feasible region determined by the system of constraints is given below:</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1732774252phpbwCTU5_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1732774252phpbwCTU5.jpeg" alt="" width="225" height="211"></picture></div></div><p>A (140, 50), B (20, 140) and C (40, 100) 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 = 3x + 3.5y</p></td><td>&nbsp;</td></tr><tr><td><p>A (140, 50)</p></td><td><p>595</p></td><td><p>Maximum</p></td></tr><tr><td><p>B (20, 140)</p></td><td><p>550</p></td><td>&nbsp;</td></tr><tr><td><p>C (40, 100)</p></td><td><p>470</p></td><td>&nbsp;</td></tr></tbody></table><p>The maximum value of z is 595 at (140, 50).</p><p>Hence, 140 bags of brand P and 50 bags of brand Q should be used to maximise the amount of nitrogen.</p><p>Thus, the maximum amount of nitrogen added to the garden is 595 kg.</p>

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