- Topic
- Related rates
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Related rates practice question 6
An online shop sells copies of a board game a week when the price is dollars, where . The price is now $25 and is being raised by $0.50 a week. At what rate is the weekly demand changing?
Give a signed rate in games per week.
Demand depends on the price, and the price changes week by week. Which rule links to , and should demand rise or fall as the price goes up?
Write and differentiate both sides with respect to . The chain rule brings in .
A signed rate, exact or to two decimal places.
Theory UBC Math 100 §5CLP-1 §3.2Key results
Worked solution
Strategy
Demand depends on the price and the price depends on time, so differentiate the demand relation with respect to time with the chain rule, then substitute the price and its rate at this instant.
Working
Power rule, then the chain rule: p depends on t.
Substitute p = 25, where p√p = 125, and dp/dt = 0.5 only now.
Answer
The answer is demand is falling at 6 games per week, a rate of −6.
Check
A week from now the price is $25.50 and demand 3000/√25.5 ≈ 594.1 games, about 6 fewer than the 600 sold now.
Where this goes wrong
Putting p = 25 into the demand relation first, which gives the constant q = 600 and a rate of 0. Differentiate while p is still a variable, then substitute.
Leans on
Negative and fractional powers, from Differentiation rules
Saved in this browser only.