type the correct answer in each box. round your answers to the nearest dollar.\nthese are the cost and…

Explanation:

Step1: Define profit function

The profit function $P(x)$ is given by $P(x)=R(x)-C(x)$. Substitute $R(x)= - 31.72x^{2}+2030x$ and $C(x)=-126.96x + 26391$ into the profit - function formula: $P(x)=(-31.72x^{2}+2030x)-(-126.96x + 26391)=-31.72x^{2}+(2030 + 126.96)x-26391=-31.72x^{2}+2156.96x-26391$.

Step2: Find the x - value of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is $x=-\frac{b}{2a}$. In the profit function $P(x)=-31.72x^{2}+2156.96x-26391$, $a=-31.72$ and $b = 2156.96$. $x=-\frac{2156.96}{2\times(-31.72)}=\frac{2156.96}{63.44}=34$.

Step3: Find the maximum profit

Substitute $x = 34$ into the profit function $P(x)$: $P(34)=-31.72\times34^{2}+2156.96\times34-26391$ $=-31.72\times1156+73336.64-26391$ $=-36668.32+73336.64-26391$ $=36668.32-26391$ $=10277.32\approx10277$.

Step4: Find the selling price

The revenue function is $R(x)=-31.72x^{2}+2030x$. The selling price per unit $p$ is given by $p=\frac{R(x)}{x}$ (when $x\neq0$). Since the maximum - profit occurs at $x = 34$, we can also use the fact that for a quadratic revenue function $R(x)=ax^{2}+bx$, the marginal revenue $R^\prime(x)=2ax + b$. At the maximum - profit point, marginal revenue equals marginal cost. Another way is to use the relationship between revenue and the number of units. The selling price per bag is $\frac{R(34)}{34}$. $R(34)=-31.72\times34^{2}+2030\times34=-36668.32 + 69020=32351.68$. The selling price per bag $p=\frac{32351.68}{34}\approx951.52\approx952$.

Answer:

10277, 952