Question

The cost, C, to produce b baseball bats per day is modeled by the function C(b) = 0.06b2 – 7.2b + 390. What number of bats should be produced to keep costs at a minimum?

A. 27 bats
B. 60 bats
C. 174 bats
D. 390 bats

Answer 1

To determine the number of bats that should be produced for the minimum cost, differentiate the given equation and equate to zero.
dC(b) / dt = (0.06)(2)b - 7.2
0 = 0.12b - 7.2
The value of b from the derived equation is 60. Therefore, the company should produced 60 bats for the minimum cost. Thus, the answer is letter B.

Answer 2

we have

`C(b)=0.06b^(2)-7.2b+390`

this is a quadratic equation (vertical parabola) open up

so

the vertex is a minimum

Convert the equation in the vertex form to find the vertex

`C(b)=0.06b^(2)-7.2b+390`

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`C(b)-390=0.06b^(2)-7.2b`

Factor the leading coefficient

`C(b)-390=0.06(b^(2)-120b)`

Complete the square. Remember to balance the equation by adding the same constants to each side

`C(b)-390+216=0.06(b^(2)-120b+3,600)`

`C(b)-174=0.06(b^(2)-120b+3,600)`

Rewrite as perfect squares

`C(b)-174=0.06(b-60)^(2)`

`C(b)=0.06(b-60)^(2)+174` --------> equation in vertex form

the vertex is the point
`(60,174)`

the vertex is the minimum of the function

so

the minimum cost is
`174` and the number of bats to keep cost at minimum is
`60`

therefore

the answer is the option

B. 60 bats