Question

Healthy food corporation is planning to produce 600 Kg of healthy food in a week combining three nutrients components X.Y and Z that cost Rs. 12 Rs 16 and Rs14 per Kg respectively, the croption can spend only Rs 9240 in a week to buy the nutrients components from the markets, the only constriant the corporation has to meet in producing 1 kg of pack healthy food is that it must contains matrient component X to he Twice of 2. Find the amount of each nutrients components which should be mixed in final impot by using Cramer's rule. ​

Answer 1

SOLUTION

GIVEN

Healthy food corporation is planning to produce 600 Kg of healthy food in a week combining three nutrients components X , Y and Z that cost Rs. 12 Rs 16 and Rs 14 per Kg respectively, the corporation can spend only Rs 9240 in a week to buy the nutrients components from the markets, the only constriant the corporation has to meet in producing 1 kg of pack healthy food is that it must contains nutrient component X to he Twice of Z .

TO DETERMINE

The amount of each nutrients components which should be mixed in final impot by using Cramer's rule.

EVALUATION

Let

The amount of component X = x kg

The amount of component Y = y kg

The amount of component Z = z kg

By the condition 1

x + y + z = 600 - - - - - - (1)

By the condition 2

12x + 16y + 14z = 9240

⇒ 6x + 8y + 7z = 4620 - - - - - - (2)

By the condition 3

x = 2z

⇒ x - 2z = 0 - - - - - - (3)

The given system of equations are

x + y + z = 600

x + 0.y - 2z = 0

6x + 8y + 7z = 4620

The given system of equations can be rewritten in Matrix form as

$\displaystyle \begin{pmatrix} 1 & 1 & 1\\ 1 & 0 & - 2 \\ 6 & 8 & 7 \end{pmatrix} \displaystyle\begin{pmatrix} \sf{x}\\ \sf{y} \\ \sf{z} \end{pmatrix} = \displaystyle\begin{pmatrix} \sf{600}\\ \sf{0} \\ \sf{4620} \end{pmatrix}$

Now we solve using Cramer's Rule

$\displaystyle \sf{D} = \begin{vmatrix} 1 & 1 & 1\\ 1 & 0 & - 2 \\ 6 & 8 & 7 \end{vmatrix} = 5$

$\displaystyle \sf{D_x} = \begin{vmatrix} 600 & 1 & 1\\ 0 & 0 & - 2 \\ 4620 & 8 & 7 \end{vmatrix} = 360$

$\displaystyle \sf{D_y} = \begin{vmatrix} 1 & 600 & 1\\ 1 & 0 & - 2 \\ 6 & 4620 & 7 \end{vmatrix} = 2460$

$\displaystyle \sf{D_z} = \begin{vmatrix} 1 & 1 & 600\\ 1 & 0 & 0 \\ 6 & 8 & 4620 \end{vmatrix} = 180$

Thus we get

$\displaystyle \sf{x = \frac{D_x}{D} } = \frac{360}{5} = 72$

$\displaystyle \sf{y = \frac{D_y}{D} } = \frac{2460}{5} = 492$

$\displaystyle \sf{z = \frac{D_x}{D} } = \frac{180}{5} = 36$

FINAL ANSWER

The amount of component X = 72 kg

The amount of component Y = 492 kg

The amount of component Z = 36 kg

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