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