Maximize Profit: Demand, Cost, and Revenue Functions | Calculus Word Problem

Published on April 12, 2026

Calculus

Learn how to maximize annual profit by analyzing demand and cost functions to find optimal production quantity and price using calculus. Solved step-by-step.

Question

1. A company estimates that the demand for its product fluctuates with price it charges. The demand function is q = 280, 000 – 400p Where q equals the number of units demanded and p equals the price in dollars. The total cost of producing q units of the product is estimated by the function C = 350, 000 + 300q + 0.0015q2 a) Determine how many units of q should be produced in order to maximize annual profit. b) What price should be charged? c) What is the annual profit expected to equal? 2. The marginal revenue function for a firm’s product is; MR = -0.04x + 10 Where x equals the number of units sold. What is the additional revenue if sales increase from 100 units to 200 units.

Answer

Here’s a step-by-step solution to the provided problems:

Problem 1: Maximizing Annual Profit

Given:

Demand function:
Total Cost function:

Goal: Maximize annual profit.

Key Concepts:

Profit § = Total Revenue ® - Total Cost ©
Total Revenue ® = Price § * Quantity (q)
To maximize profit, we need to find the quantity (q) where the derivative of the profit function with respect to q is zero () and the second derivative is negative ().

Step 1: Express Revenue ® in terms of q.
From the demand function, we can express price § in terms of quantity (q):

Now, we can express Total Revenue ® as a function of q:

Step 2: Express Profit § in terms of q.

Step 3: Find the quantity (q) that maximizes profit.
To find the maximum profit, we take the derivative of the profit function with respect to q and set it to zero:

Set

To confirm this is a maximum, we check the second derivative:

Since

, the profit is indeed maximized at

a) Determine how many units of q should be produced in order to maximize annual profit.
The number of units to be produced to maximize annual profit is 50,000 units.

b) What price should be charged?
We use the price function we derived earlier, substituting

The price that should be charged is $575.

c) What is the annual profit expected to equal?
We substitute

into the profit function:

The annual profit expected to equal $9,650,000.

Problem 2: Additional Revenue from Increased Sales

Given:

Marginal Revenue function:
- Where is the number of units sold.

Goal: Find the additional revenue if sales increase from 100 units to 200 units.

Key Concept:
Marginal Revenue (MR) is the derivative of the Total Revenue ® function with respect to the number of units (x). Therefore, to find the additional revenue, we need to integrate the MR function over the given range of units.

Step 1: Set up the integral.
The additional revenue from increasing sales from

is given by:

In this case,

and

Step 2: Evaluate the integral.

Now, substitute the upper and lower limits:

The additional revenue if sales increase from 100 units to 200 units is $400.

Theorems

• Profit Maximization Principle

Profit is maximized when the derivative of the profit function with respect to quantity is zero, i.e.,

, and the second derivative is negative, i.e.,

. This corresponds to the point where marginal revenue (

) equals marginal cost (

). For a firm with revenue

and cost

, profit

. Thus,

. Setting

implies

• Fundamental Theorem of Calculus (Part 2)

The definite integral of a rate of change of a quantity gives the total change in that quantity over an interval. If

is the derivative of

(i.e.,

), then the integral of

from

is the difference in

at these points:

. In the context of revenue, if

is the marginal revenue function, then the total change in revenue (additional revenue) from selling

units is

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