22. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

Let the diet contain x and y packets of foods P and Q, respectively. Hence,

x ≥ 0 and y ≥ 0

The mathematical formulation of the given problem is given below:

$Maximisez=6x+3y\dots \dots \dots \dots ..\left(i\right)$

Subject to the constraints,

$\begin{array}{}\end{array}4x+y\ge 80\dots \dots \dots \dots .\left(ii\right)$

$x+5y\ge 115\dots \dots \dots .\left(iii\right)$

$3x+2y\le 150\dots \dots \dots \dots \left(iv\right)$

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

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

A (15, 20), B (40, 15) and C (2, 72) are the corner points of the feasible region

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

Corner Point	z = 6x + 3y
A (15, 20)	150
B (40, 15)	285	Maximum
C (2, 72)	228

So, the maximum value of z is 285 at (40, 15).

Hence, to maximise the amount of vitamin A in the diet, 40 packets of food P and 15 packets of food Q should be used.

The maximum amount of vitamin A in the diet is 285 units.

<p>Let the diet contain x and y packets of foods P and Q, respectively. Hence, </p><p>x ≥ 0 and y ≥ 0</p><p>The mathematical formulation of the given problem is 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>6</mn><mi>x</mi><mo>+</mo><mn>3</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>4</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mn>8</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>5</mn><mi>y</mi><mo>≥</mo><mn>1</mn><mn>1</mn><mn>5</mn><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>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>≤</mo><mn>1</mn><mn>5</mn><mn>0</mn><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/1732774038phpMJFuUr_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1732774038phpMJFuUr.jpeg" alt="" width="315" height="231"></picture></div></div><p>A (15, 20), B (40, 15) and C (2, 72) are the corner points of the feasible region</p><p>The values of z at these corner points are as given below:</p><table><tbody><tr><td><p>Corner Point</p></td><td><p>z = 6x + 3y</p></td><td>&nbsp;</td></tr><tr><td><p>A (15, 20)</p></td><td><p>150</p></td><td>&nbsp;</td></tr><tr><td><p>B (40, 15)</p></td><td><p>285</p></td><td><p>Maximum</p></td></tr><tr><td><p>C (2, 72)</p></td><td><p>228</p></td><td>&nbsp;</td></tr></tbody></table><p>So, the maximum value of z is 285 at (40, 15).</p><p>Hence, to maximise the amount of vitamin A in the diet, 40 packets of food P and 15 packets of food Q should be used.</p><p>The maximum amount of vitamin A in the diet is 285 units.</p>

0 Upvotes 0 Upvotes 0 Downvotes

• • Report Abuse
• Comment