glo4veryndasa
contestada

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

Respuesta :

meerkat18
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. 
calculista

we have

[tex]C(b)=0.06b^{2}-7.2b+390[/tex]

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

[tex]C(b)=0.06b^{2}-7.2b+390[/tex]

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

[tex]C(b)-390=0.06b^{2}-7.2b[/tex]

Factor the leading coefficient

[tex]C(b)-390=0.06(b^{2}-120b)[/tex]

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

[tex]C(b)-390+216=0.06(b^{2}-120b+3,600)[/tex]

[tex]C(b)-174=0.06(b^{2}-120b+3,600)[/tex]

Rewrite as perfect squares

[tex]C(b)-174=0.06(b-60)^{2}[/tex]

[tex]C(b)=0.06(b-60)^{2}+174[/tex] --------> equation in vertex form

the vertex is the point [tex](60,174)[/tex]

the vertex is the minimum of the function

so

the minimum cost is [tex]174[/tex] and the number of bats to keep cost at minimum is [tex]60[/tex]

therefore

the answer is the option

B. 60 bats


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