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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?

A27 bats
B60 bats
C174 bats
D390 bats

Verified Answer :

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

\(C(b)=0.06b^{2}-7.2b+390\)

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


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