- Topic
- Related rates
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Related rates practice question 12
A theatre sells tickets a week when the price is dollars per ticket, where . The price is being raised at $0.40 per week. How fast is the weekly revenue changing at the moment the price is $30?
Give a signed rate in dollars per week.
Raising the price earns more on each ticket but sells fewer tickets. How is the weekly revenue built from and , and does either factor stay fixed?
Revenue is price times number sold, , and both and change with time.
Product rule: . Differentiating the demand relation gives .
A number, negative if the revenue is falling.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Revenue is a product of two quantities that both change with the price. Differentiate R = pq with the product rule, get dq/dt from the demand relation, and substitute the instant.
Working
Product rule: the price and the number sold both change.
Values at this instant, from the demand relation and its derivative.
The higher price brings in $1000 a week more, the lost sales cost $600 a week, and the net is a gain.
Answer
The answer is revenue is rising at 400 dollars per week.
Check
A week later the price is $30.40, 2480 tickets sell and revenue is $75,392, up $392 on the $75,000 of now, close to $400.
Where this goes wrong
Holding one factor of R = pq fixed. Keeping the 2500 tickets fixed gives 1000 and keeping the $30 price fixed gives −600; the true rate is their sum, 400, because the price and the number sold both change.
Leans on
The product rule, from Differentiation rules
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