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