Question

the total cost of producing a type of truck is given by ( c(x) = 23000 - 60x + 0.05x^{2} ), where ( x ) is the number of trucks produced. how many trucks should be produced to incur minimum cost? answer/how to enter your answer (opens in new window) 2 points keyboard keyboard shortcuts trucks

Explanation:

Step1: Find the derivative of the cost function

The cost function is \( C(x) = 23000-60x + 0.05x^{2}\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(C^\prime(x)=-60 + 0.1x\).

Step2: Set the derivative equal to zero

To find the critical points, set \(C^\prime(x)=0\).
So, \(-60+0.1x = 0\).

Step3: Solve for \(x\)

Add \(60\) to both sides: \(0.1x=60\).
Divide both sides by \(0.1\): \(x=\frac{60}{0.1}=600\).

Step4: Check the second - derivative

The second - derivative \(C^{\prime\prime}(x)=0.1>0\). Since the second - derivative is positive, the function \(C(x)\) has a minimum at \(x = 600\).

Answer:

\(600\)