Question

when a management training company prices its seminar on management techniques at $400 per person, 1,000 people will attend the seminar. the company estimates that for each $5 reduction in price, an additional 20 people will attend the seminar. what is the first derivative of revenue?

Answer 1

The first derivative of revenue with respect to price for the management training company's seminar is calculated using the product rule and is represented by the formula R'(x) = 3,000 - 200x.

Step-by-step explanation:

The question is asking for the first derivative of revenue with respect to the price.

As the management training company reduces the seminar price by $5, it gains an additional 20 attendees per reduction. Let's denote the number of $5 reductions by x, the initial number of attendees as A (1,000), and the initial price as P ($400).

The total revenue (R) as a function of x is R(x) = (P - 5x)(A + 20x).

To find the first derivative of revenue, R'(x), apply the product rule: R'(x) = P'(x)(A + 20x) + (P - 5x)(A + 20x)'.

Substituting P = $400 and A = 1,000, we get R'(x) = -5(1,000 + 20x) + (400 - 5x)(20)

which simplifies to R'(x) = -5,000 - 100x + 8,000 - 100x, and further to R'(x) = 3,000 - 200x.

This derivative represents the rate of change of revenue with respect to the number of $5 price reductions.