Question

1. the cost, in hundreds, to produce x thousand mechanical pencils is represented by the function ( c(x)=x^{2}-10x + 30 ).

a. how many pencils should be produced to minimize the cost?
b. what is the minimum cost?

Explanation:

Step1: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\), the \(x\)-coordinate of the vertex is given by \(x=-\frac{b}{2a}\).
In the function \(C(x)=x^{2}-10x + 30\), \(a = 1\), \(b=-10\).
So, \(x=-\frac{-10}{2\times1}=5\).

Step2: Calculate the minimum cost

Substitute \(x = 5\) into the cost function \(C(x)\).
\(C(5)=5^{2}-10\times5 + 30\)
\(=25-50 + 30\)
\(=5\)

Answer:

A. \(5000\) pencils should be produced to minimize the cost.
B. The minimum cost is \(500\).