Question Type 2: Optimizing profit or cost functions under a given context Exercises

A firm faces a linear demand $p(q) = 100 - 2q$ and incurs cost $C(q) = 20q + 50$. Find the output level $q$ that maximizes profit.

The revenue for a product is given by $R(q) = 40q - 0.5q^2$ and the cost is given by $C(q) = 5q + 100$, where $q$ is the quantity produced. Find the value of $q$ that maximizes the profit.

A company sells a product at a fixed price of $18$ per unit and has cost function $C(q) = 5q^2 - 22q + 200$. Determine the quantity $q$ that maximizes profit.

A product’s price depends on demand, $p(q) = 200 - q$, and its cost is $C(q) = 0.01q^2 + 2q + 500$. Determine the output level $q$ that maximizes profit.

A company’s revenue is $R(q)=150\ln(q)$ and its cost is $C(q)=20q+100$. Find the output $q$ that maximizes profit.

A bakery sells cakes at $p(q)=60-0.5q$ dollars each and has cost $C(q)=10q+150$. Find the production level $q$ that maximizes profit.

A smartphone maker earns $300$ dollars per unit and has cost function $C(q) = 0.5q^2 + 50q + 1000$. Determine the output $q$ that maximizes profit.

A car manufacturer faces demand $p(q) = 50\,000 - 0.01q$ and cost $C(q) = 20\,000q + 10^6$. Find the output $q$ that maximizes profit.

A service provider has profit function $P(q) = 70q - 0.005q^2 - 200$, where $q$ is the quantity of units sold. Find the value of $q$ that maximizes profit.

The price-demand function is $p(q)=80-0.1q$ and the cost function is $C(q)=2q+300$. Determine the value of $q$ that maximizes profit.

Total revenue is $R(q)=120q-0.2q^2$ and cost is $C(q)=30q+400$. Determine the output level $q \ge 0$ that maximizes profit.

The total cost of producing $q$ units is given by $C(q) = q^3 - 30q^2 + 300q + 200$.

Determine the output level $q$ that minimizes the marginal cost.