To determine the number of pennies in the stack using Cramer's Rule, let [tex]\( p \)[/tex] be the number of pennies, [tex]\( n \)[/tex] the number of nickels, and [tex]\( d \)[/tex] the number of dimes. We form the following system of linear equations based on the given conditions:
1. The total number of coins:
[tex]\[ p + n + d = 24 \][/tex]
2. The total value of the coins (in dollars):
[tex]\[ 0.01p + 0.05n + 0.10d = 1.16 \][/tex]
3. The total height of the coins (in mm):
[tex]\[ 1.52p + 1.95n + 1.35d = 37.27 \][/tex]
These equations can be written in matrix form as:
[tex]\[
\begin{cases}
p + n + d = 24 \\
0.01p + 0.05n + 0.10d = 1.16 \\
1.52p + 1.95n + 1.35d = 37.27
\end{cases}
\][/tex]
This can be represented as:
[tex]\[ A \vec{x} = \vec{B} \][/tex]
where
[tex]\[
A = \begin{bmatrix}
1 & 1 & 1 \\
0.01 & 0.05 & 0.10 \\
1.52 & 1.95 & 1.35
\end{bmatrix},
\quad
\vec{x} = \begin{bmatrix}
p \\
n \\
d
\end{bmatrix},
\quad
\vec{B} = \begin{bmatrix}
24 \\
1.16 \\
37.27
\end{bmatrix}
\][/tex]
To use Cramer's Rule, we first find the determinant of matrix [tex]\( A \)[/tex], denoted as [tex]\( \text{det}(A) \)[/tex]:
[tex]\[
\text{det}(A) = \begin{vmatrix}
1 & 1 & 1 \\
0.01 & 0.05 & 0.10 \\
1.52 & 1.95 & 1.35
\end{vmatrix}
= -0.0455
\][/tex]
Next, to find the determinant for the matrix [tex]\( A_p \)[/tex] where the first column (corresponding to [tex]\( p \)[/tex]) is replaced by the constants from [tex]\( \vec{B} \)[/tex]:
[tex]\[
A_p = \begin{bmatrix}
24 & 1 & 1 \\
1.16 & 0.05 & 0.10 \\
37.27 & 1.95 & 1.35
\end{bmatrix}
\][/tex]
We calculate the determinant of [tex]\( A_p \)[/tex]:
[tex]\[
\text{det}(A_p) = \begin{vmatrix}
24 & 1 & 1 \\
1.16 & 0.05 & 0.10 \\
37.27 & 1.95 & 1.35
\end{vmatrix}
= -0.5005
\][/tex]
Using Cramer's Rule, the number of pennies [tex]\( p \)[/tex] is given by:
[tex]\[
p = \frac{\text{det}(A_p)}{\text{det}(A)} = \frac{-0.5005}{-0.0455} = 11
\][/tex]
Therefore, there are 11 pennies in the stack.
