To determine the number of baseball bats that should be produced to minimize the cost, we need to analyze the given cost function. The cost function provided is:
[tex]\[ C(b) = 0.06b^2 - 7.2b + 390 \][/tex]
This is a quadratic function of the form [tex]\( C(b) = ab^2 + bb + c \)[/tex], where:
- [tex]\( a = 0.06 \)[/tex]
- [tex]\( b = -7.2 \)[/tex]
- [tex]\( c = 390 \)[/tex]
For a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], the vertex (which represents the minimum point in this case, since [tex]\( a > 0 \)[/tex]) can be found using the vertex formula:
[tex]\[ b = -\frac{B}{2A} \][/tex]
Here, [tex]\( A \)[/tex] and [tex]\( B \)[/tex] correspond to the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] from the quadratic function. Plugging in our values:
- [tex]\( A = 0.06 \)[/tex]
- [tex]\( B = -7.2 \)[/tex]
The number of baseball bats [tex]\( b \)[/tex] that minimizes the cost is:
[tex]\[ b = -\frac{-7.2}{2 \cdot 0.06} \][/tex]
Simplify the expression:
[tex]\[ b = \frac{7.2}{2 \cdot 0.06} \][/tex]
[tex]\[ b = \frac{7.2}{0.12} \][/tex]
[tex]\[ b = 60 \][/tex]
Therefore, the number of baseball bats that should be produced to minimize the cost is:
[tex]\[ \boxed{60} \][/tex]
So, the correct answer is [tex]\( 60 \)[/tex] bats.
