Question

the total cost of producing a type of boat is given by c(x) = 16000 - 80x + 0.1x², where x is the number of boats produced. how many boats should be produced to incur minimum cost?

Explanation:

Step1: Identify the function type

The cost function $C(x)=16000 - 80x+0.1x^{2}$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a = 0.1$, $b=-80$, $c = 16000$.

Step2: Use the formula for the vertex of a quadratic

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the minimum for $a>0$) is $x=-\frac{b}{2a}$.
Substitute $a = 0.1$ and $b=-80$ into the formula: $x=-\frac{-80}{2\times0.1}$.

Step3: Calculate the value of x

$x=\frac{80}{0.2}=400$.

Answer:

400