a company produces and sells a product. its profit function is given by $p(q)=r(q)-c(q)$ where $r(q)$ is the…

a company produces and sells a product. its profit function is given by $p(q)=r(q)-c(q)$ where $r(q)$ is the revenue function and $c(q)$ is the cost function, both in terms of quantity $q$. at a quantity of $q = 105$, the rate of change of revenue is $r(105)=10$ and the rate of change of cost is $c(105)= - 13$. what is $p(105)$?

a company produces and sells a product. its profit function is given by $p(q)=r(q)-c(q)$ where $r(q)$ is the revenue function and $c(q)$ is the cost function, both in terms of quantity $q$. at a quantity of $q = 105$, the rate of change of revenue is $r(105)=10$ and the rate of change of cost is $c(105)= - 13$. what is $p(105)$?

Answer

Explanation:

Step1: Differentiate profit function

By the difference - rule of differentiation, if $P(q)=R(q)-C(q)$, then $P^{\prime}(q)=R^{\prime}(q)-C^{\prime}(q)$.

Step2: Substitute given values

We are given that $q = 105$, $R^{\prime}(105)=10$ and $C^{\prime}(105)=- 13$. Substitute these values into the formula for $P^{\prime}(q)$. So $P^{\prime}(105)=R^{\prime}(105)-C^{\prime}(105)$. $P^{\prime}(105)=10-(-13)$.

Step3: Simplify the expression

$P^{\prime}(105)=10 + 13=23$.

Answer:

$23$