Question

we are modeling the profit in dollars for every q - units sold of a product. if $r(q)=\frac{100q^{2}-30}{2}$ and $c(q)=30q + 20$. calculate the marginal profit when q = 4. question 8 in the previous problem, what is the unit of measurement that marginal profit is expressed as? dollars/unit sold dollars units sold units sold/dollar earned

Explanation:

Step1: Define profit function

The profit function $P(q)$ is $P(q)=R(q)-C(q)$. Given $R(q)=\frac{100q^{2}-30}{2} = 50q^{2}-15$ and $C(q)=30q + 20$, then $P(q)=50q^{2}-30q-35$.

Step2: Find marginal - profit function

The marginal - profit function $P'(q)$ is the derivative of the profit function. Using the power rule $\frac{d}{dq}(ax^{n})=nax^{n - 1}$, we have $P'(q)=\frac{d}{dq}(50q^{2}-30q - 35)=100q-30$.

Step3: Evaluate marginal - profit at $q = 4$

Substitute $q = 4$ into $P'(q)$: $P'(4)=100\times4-30=400 - 30=370$.

Marginal profit represents the change in profit for a one - unit change in the quantity sold. Since profit is in dollars and quantity is in units sold, the unit of marginal profit is dollars per unit sold.

Answer:

370