Question

section 2.8: optimization (homework)
score: 30/120 answered: 3/12
progress saved done
question 4
0/10 pts 4 99 details
the cost, in dollars, to produce x designer dog leashes is c(x)=5x + 1, and the revenue function, in dollars, is r(x)=-4x^{2}+141x
find the profit function.
p(x)=
find the number of leashes which need to be sold to maximize the profit.
select an answer
find the maximum profit.
select an answer
find the price to charge per leash to maximize profit.
select an answer
what would be the best reasons to either pay or not pay that much for a leash?

Explanation:

Step1: Recall profit - revenue - cost formula

The profit function $P(x)$ is given by $P(x)=R(x)-C(x)$. Given $C(x) = 5x + 1$ and $R(x)=-4x^{2}+141x$, we substitute these functions into the profit - formula.
$P(x)=(-4x^{2}+141x)-(5x + 1)=-4x^{2}+141x - 5x-1=-4x^{2}+136x - 1$.

Step2: Find the number of units to maximize profit

For a quadratic function $y = ax^{2}+bx + c$ ($a
eq0$), the $x$ - value of the vertex (which gives the maximum or minimum) is $x=-\frac{b}{2a}$. In the profit function $P(x)=-4x^{2}+136x - 1$, $a=-4$ and $b = 136$.
$x=-\frac{136}{2\times(-4)}=\frac{136}{8}=17$.

Step3: Find the maximum profit

Substitute $x = 17$ into the profit function $P(x)=-4x^{2}+136x - 1$.
$P(17)=-4\times(17)^{2}+136\times17-1=-4\times289 + 2312-1=-1156+2312 - 1=1155$.

Step4: Find the price per unit

The revenue function is $R(x)=p\times x$, where $p$ is the price per unit. We know $R(x)=-4x^{2}+141x$ and $x = 17$. Also, $R(17)=p\times17$. First, find $R(17)=-4\times(17)^{2}+141\times17=-4\times289+2397=-1156 + 2397 = 1241$. Then, $p=\frac{R(17)}{17}=\frac{1241}{17}=73$.

Answer:

$P(x)=-4x^{2}+136x - 1$
17
1155
73