Answer:
Number of bats should be produced to keep costs at a minimum is 60.
Step-by-step explanation:
Given : The cost, C, to produce b baseball bats per day is modeled by the function \(C(b) = 0.06b^2 -7.2b + 390\).
To find : What number of bats should be produced to keep costs at a minimum?
Solution :
Function in quadratic form is \(C(b) = 0.06b^2 -7.2b + 390\)
To determine the minimum point we apply the formula of quadratic equation \(ax^2+bx+c=0\) is \(x=-\frac{b}{2a}\)
On comparing with given model, a=0.06 , b=-7.2 , c=390
The minimum point is at \(b=-\frac{-7.2}{2(0.06)}\)
\(b=-\frac{-7.2}{0.12}\)
\(b=\frac{7.2}{0.12}\)
\(b=60\)
i.e. The minimum number of bats per day is 60.
The cost at b=60 is
\(C(60) = 0.06(60)^2 -7.2(60) + 390\)
\(C(60) = 216 -432 + 390\)
\(C(60) =174\)
The minimum cost is $174 and minimum number of bats is 60.
We can also determine through graph.
Refer the attached figure below.
