Sure, let’s start by defining variables for the prices of each coffee size:
- Let [tex]\( x \)[/tex] be the price of a small (bajo) coffee.
- Let [tex]\( y \)[/tex] be the price of a medium (mediano) coffee.
- Let [tex]\( z \)[/tex] be the price of a large (grande) coffee.
Given the sales data across different time slots, we can set up the following system of equations based on the earnings:
1. For the time [tex]\( 5-10 \)[/tex]:
[tex]\[ 55x + 53y + 54z = 294.12 \][/tex]
2. For the time [tex]\( 10-3 \)[/tex]:
[tex]\[ 17x + 22y + 25z = 117.89 \][/tex]
3. For the time [tex]\( 3-8 \)[/tex]:
[tex]\[ 30x + 23y + 29z = 148.87 \][/tex]
To solve this system of equations, we can use methods such as substitution, elimination, or matrix operations (e.g., Gaussian elimination). Here, we will use matrix operations.
Represent the system in matrix form [tex]\( A \cdot P = B \)[/tex], where [tex]\( A \)[/tex] is the coefficient matrix, [tex]\( P \)[/tex] is the price vector [tex]\([x, y, z]\)[/tex], and [tex]\( B \)[/tex] is the earnings vector:
[tex]\[
A = \begin{bmatrix}
55 & 53 & 54 \\
17 & 22 & 25 \\
30 & 23 & 29
\end{bmatrix}
,
P = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
,
B = \begin{bmatrix}
294.12 \\
117.89 \\
148.87
\end{bmatrix}
\][/tex]
To find [tex]\( P \)[/tex], we solve [tex]\( A \cdot P = B \)[/tex]:
[tex]\[
P = A^{-1} \cdot B
\][/tex]
The solution to the system gives us the exact prices for each coffee size:
[tex]\[
x \approx 1.62
\][/tex]
[tex]\[
y \approx 1.80
\][/tex]
[tex]\[
z \approx 2.03
\][/tex]
Now let's check each given price option to determine which one is the closest match:
1. [tex]\( \text{bajo: } \$1.60, \text{ mediano: } \$1.82, \text{ grande: } \$2.00 \)[/tex]
[tex]\[ \text{Difference} = |1.62 - 1.60| + |1.80 - 1.82| + |2.03 - 2.00| = 0.02 + 0.02 + 0.03 = 0.07 \][/tex]
2. [tex]\( \text{bajo: } \$1.61, \text{ mediano: } \$1.77, \text{ grande: } \$2.03 \)[/tex]
[tex]\[ \text{Difference} = |1.62 - 1.61| + |1.80 - 1.77| + |2.03 - 2.03| = 0.01 + 0.03 + 0 = 0.04 \][/tex]
3. [tex]\( \text{bajo: } \$1.62, \text{ mediano: } \$1.80, \text{ grande: } \$2.03 \)[/tex]
[tex]\[ \text{Difference} = |1.62 - 1.62| + |1.80 - 1.80| + |2.03 - 2.03| = 0 + 0 + 0 = 0 \][/tex]
4. [tex]\( \text{bajo: } \$1.65, \text{ mediano: } \$1.79, \text{ grande: } \$2.07 \)[/tex]
[tex]\[ \text{Difference} = |1.62 - 1.65| + |1.80 - 1.79| + |2.03 - 2.07| = 0.03 + 0.01 + 0.04 = 0.08 \][/tex]
The smallest difference is found in option 3:
[tex]\[
\text{bajo: }\$1.62, \text{ mediano: }\$1.80, \text{ grande: }\$2.03
\][/tex]
Therefore, the closest match is option 3. So, the prices of each regular coffee size are:
- Small (bajo): \[tex]$1.62
- Medium (mediano): \$[/tex]1.80
- Large (grande): \$2.03
The correct option is number 3.
