To determine the calorie content per gram for fat, carbohydrates, and protein, we set up a system of linear equations based on the given nutritional content table and solve it using matrix equations.
The nutritional content table is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline & \text{Fat (g)} & \text{Carb (g)} & \text{Protein (g)} & \text{Calories} \\
\hline \text{Burger} & 29 & 46 & 25 & 545 \\
\hline \text{Fries} & 25 & 63 & 6 & 501 \\
\hline \text{Shake} & 21 & 115 & 15 & 709 \\
\hline
\end{array}
\][/tex]
We denote:
- [tex]\( f \)[/tex] as the calories per gram of fat
- [tex]\( c \)[/tex] as the calories per gram of carbohydrates
- [tex]\( p \)[/tex] as the calories per gram of protein
The total calories for each item can be expressed as the sum of the calories from fat, carbohydrates, and protein. Thus, we have the following system of linear equations:
1. [tex]\( 29f + 46c + 25p = 545 \)[/tex] (Burger)
2. [tex]\( 25f + 63c + 6p = 501 \)[/tex] (Fries)
3. [tex]\( 21f + 115c + 15p = 709 \)[/tex] (Shake)
This system of linear equations can be represented in matrix form as:
[tex]\[ A \mathbf{x} = \mathbf{B} \][/tex]
Where:
[tex]\[ A = \begin{pmatrix}
29 & 46 & 25 \\
25 & 63 & 6 \\
21 & 115 & 15
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
f \\
c \\
p
\end{pmatrix}, \quad
\mathbf{B} = \begin{pmatrix}
545 \\
501 \\
709
\end{pmatrix} \][/tex]
To solve for [tex]\(\mathbf{x}\)[/tex], we find [tex]\( \mathbf{x} = A^{-1} \mathbf{B} \)[/tex].
By solving this system of equations, we find the values for [tex]\( f \)[/tex], [tex]\( c \)[/tex], and [tex]\( p \)[/tex]:
[tex]\[
\begin{pmatrix}
f \\
c \\
p
\end{pmatrix} = \begin{pmatrix}
9.0 \\
4.0 \\
4.0
\end{pmatrix}
\][/tex]
Thus, the calories per gram for each component are:
[tex]\[
\text{Calories per gram:}
\][/tex]
[tex]\[
\text{Fat: } \boxed{9.0}
\][/tex]
[tex]\[
\text{Carbohydrate: } \boxed{4.0}
\][/tex]
[tex]\[
\text{Protein: } \boxed{4.0}
\][/tex]
So, there are 9.0 calories per gram of fat, 4.0 calories per gram of carbohydrates, and 4.0 calories per gram of protein.
