To determine the number of baseball bats [tex]\( b \)[/tex] that should be produced to minimize the cost [tex]\( C \)[/tex] described by the function [tex]\( C(b) = 0.06b^2 - 7.2b + 390 \)[/tex], we need to find the vertex of the quadratic function.
In a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the vertex, which provides either the maximum or the minimum value of the function, is found at the value of [tex]\( x \)[/tex] given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In the given cost function [tex]\( C(b) = 0.06b^2 - 7.2b + 390 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is 0.06
- The coefficient [tex]\( b \)[/tex] is -7.2
Substitute these values into the vertex formula:
[tex]\[ b = -\frac{-7.2}{2 \times 0.06} \][/tex]
[tex]\[ b = \frac{7.2}{0.12} \][/tex]
Calculating this yields:
[tex]\[ b \approx 60 \][/tex]
Thus, the number of bats that should be produced to keep costs at a minimum is 60.
Therefore, the correct answer is:
60 bats
