the function $c(x)=600x - 0.3x^{2}$ represents total costs for a company to produce a product, where $c$ is…

the function $c(x)=600x - 0.3x^{2}$ represents total costs for a company to produce a product, where $c$ is the total cost in dollars and $x$ is the number of units sold. what number of units would produce a maximum cost? what is the maximum cost?

the function $c(x)=600x - 0.3x^{2}$ represents total costs for a company to produce a product, where $c$ is the total cost in dollars and $x$ is the number of units sold. what number of units would produce a maximum cost? what is the maximum cost?

Answer

Answer:

  1. Number of units for maximum cost: 1000
  2. Maximum cost: 300000

Explanation:

Step1: Identify the function type

The cost function $C(x)=600x - 0.3x^{2}$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=- 0.3$, $b = 600$, and $c = 0$.

Step2: Find the x - value for the vertex

For a quadratic function $y=ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the maximum or minimum value) is $x=-\frac{b}{2a}$. Substituting $a=-0.3$ and $b = 600$ into the formula: [x=-\frac{600}{2\times(-0.3)}=\frac{600}{0.6}=1000]

Step3: Find the maximum cost

Substitute $x = 1000$ into the cost function $C(x)=600x-0.3x^{2}$. [C(1000)=600\times1000-0.3\times(1000)^{2}] [=600000 - 0.3\times1000000] [=600000-300000] [=300000]