the function $c(x)=400x - 0.2x^{2}$ represents the total costs for a company to produce a product, where $c$…
the function $c(x)=400x - 0.2x^{2}$ represents the 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. which statement is true?\none thousand units have the minimum cost of $200,000.\none thousand units have the maximum cost of $200,000.\ntwo thousand units have the minimum cost of $800,000.\ntwo thousand units have the maximum cost of $800,000.
Answer
Explanation:
Step1: Identificar la forma de la función
La función $C(x)=400x - 0.2x^{2}$ es una función cuadrática de la forma $y = ax^{2}+bx + c$, donde $a=- 0.2$, $b = 400$ y $c = 0$. Como $a=-0.2<0$, la parábola se abre hacia abajo y tiene un máximo.
Step2: Encontrar el valor de $x$ del vértice
El valor de $x$ del vértice de una función cuadrática $y = ax^{2}+bx + c$ se da por la fórmula $x=-\frac{b}{2a}$. Sustituyendo $a=-0.2$ y $b = 400$: [x=-\frac{400}{2\times(-0.2)}=\frac{-400}{-0.4}=1000]
Step3: Encontrar el valor máximo de la función
Sustituir $x = 1000$ en la función $C(x)=400x-0.2x^{2}$: [C(1000)=400\times1000-0.2\times(1000)^{2}] [C(1000)=400000 - 0.2\times1000000] [C(1000)=400000-200000] [C(1000)=200000]
Answer:
One thousand units have the maximum cost of $200,000.