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February 19th, 2017, 06:06 PM   #1
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Help Solving this Matrix Word Problem???

I need help with this and have no idea where to start. Can someone walk me through how to do this?

A nutritionist planning a diet for a rugby player wants him to consume 3,650 Calories and 650 grams of food daily. Calories from carbohydrates and fat will be 70% of the total Calories. There are 4, 4, and 9 Calories per gram for protein, carbohydrates, and fat, respectively. Find the number of daily grams of fat the diet will need to include by writing a system of equations, and then solving it using Cramer's Rule.
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February 19th, 2017, 06:33 PM   #2
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The first thing you need to do is write down a system of equations. As there will be three unknowns (the amount of protien, carbohydrates and fat) you will need to generate three equations.

What information can you use that is given in the question to help come up with these equations?
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February 19th, 2017, 10:38 PM   #3
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the unknowns are the grams of protein, carbs, and fat included in the diet.

let these be denoted by $p,~c,~f$ respectively

From the first sentence of the problem we have

$p + c + f = 650$

$4 p + 4 c + 9 f = 3650$

From the 2nd sentence

$4 c + 9 f = (0.7) 3650$

we write this as a matrix equation as

$\begin{pmatrix}1 &1 &1 \\ 4 &4 &9 \\ 0 &4 &9 \end{pmatrix} \begin{pmatrix}p \\ c \\ f \end{pmatrix} = \begin{pmatrix} 650 \\ 3650 \\ 2555 \end{pmatrix}$

To use Cramer's rule first we need the determinant of the the matrix on the left hand side of this equation.

$D =\left | \begin{pmatrix}1 &1 &1 \\ 4 &4 &9 \\ 0 &4 &9 \end{pmatrix} \right| $

then for example to find the value of $p$ we substitute the right hand side vector for the first column of the matrix on the left, and find the determinant of the resulting matrix.

$D_x =\left | \begin{pmatrix}650 &1 &1 \\ 3650 &4 &9 \\ 2555 &4 &9 \end{pmatrix} \right | $

$x = \dfrac{D_x}{D}$

Similarly

$D_y = \left | \begin{pmatrix}1 &650 &1 \\ 4 &3650 &9 \\ 0 &2555 &9 \end{pmatrix} \right |$

$y = \dfrac{D_y}{D}$

$D_z = \left |\begin{pmatrix}1 &1 &650 \\ 4 &4 &3650 \\ 0 &4 &2555 \end{pmatrix} \right |$

$z = \dfrac{D_z}{D}$

I leave it to you to calculate all these determinants.
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February 19th, 2017, 11:30 PM   #4
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$D=1[36-36]-1[36-0]+1[16-0]$

$=-36+16=-20$

$D_x=650[36-36]-1[3650×9-2555×9]+1[3650×4-2555×4]$

$=0-9[3650-2555]+4[3650-2555]$

$=-9×1095+4×1095$

$=1095[-9+4]=1095×(-5)=-5475$

$x=\dfrac{D_x}{D}=\dfrac{-5475}{-20}=\dfrac{5475}{20}=273.75$

$D_y=1[3650×9-2555×9]-650[36-0]+1[4×2555-0]$

$=9×1095-650×36+2555×4$

$=-3325$

$y=\dfrac{D_y}{D}=\dfrac{-3325}{-20}=\dfrac{3325}{20}=166.25$

$D_z=1[4×2555-4×3650]-1[4×2555-0]+650[16-0]$

$=-4200$

$z=\dfrac{D_z}{D}=\dfrac{-4200}{-20}=210$
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February 20th, 2017, 03:24 PM   #5
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Thanks you so much romsek! I was wayy to overwhelmed cause I wasn't walked through a word problem like this in class. This really helped!

Last edited by Trajamo; February 20th, 2017 at 03:54 PM.
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