question 5\nthe cost per unit c, in dollars, of producing x off - road 4x4 atv at a local manufacturing…

question 5\nthe cost per unit c, in dollars, of producing x off - road 4x4 atv at a local manufacturing plant is given by the function:\n$c(x)=2x^{2}-768x + 78528$.\na) find the number of off - road 4x4 atv that must be manufactured to minimize cost per unit.\nthe number of off - road 4x4 atv to manufacture to minimize unit cost is \n\nb) the minimum cost is $ per unit. (no dollar signs or commas.)

question 5\nthe cost per unit c, in dollars, of producing x off - road 4x4 atv at a local manufacturing plant is given by the function:\n$c(x)=2x^{2}-768x + 78528$.\na) find the number of off - road 4x4 atv that must be manufactured to minimize cost per unit.\nthe number of off - road 4x4 atv to manufacture to minimize unit cost is \n\nb) the minimum cost is $ per unit. (no dollar signs or commas.)

Answer

Explanation:

Step1: Identify the function type

The cost - function $C(x)=2x^{2}-768x + 78528$ is a quadratic function of the form $y = ax^{2}+bx + c$, where $a = 2$, $b=-768$, and $c = 78528$.

Step2: Find the x - value of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the value of $x$ that minimizes the function when $a>0$) is given by $x=-\frac{b}{2a}$. Substitute $a = 2$ and $b=-768$ into the formula: $x=-\frac{-768}{2\times2}=\frac{768}{4}=192$.

Step3: Find the minimum cost

Substitute $x = 192$ into the cost function $C(x)=2x^{2}-768x + 78528$. $C(192)=2\times(192)^{2}-768\times192 + 78528$. First, calculate $2\times(192)^{2}=2\times36864 = 73728$. Second, calculate $768\times192=147456$. Then $C(192)=73728-147456 + 78528$. $C(192)=73728+78528-147456=4800$.

Answer:

a) 192 b) 4800