- Topic
- Related rates
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Related rates practice question 26
A snowball melts so that its radius decreases at 0.5 cm per minute. A student finds how fast its volume is changing when the radius is 6 cm, as shown. Find the first line that is wrong.
Check each line in turn: is the formula right, is the derivative right, and does each value put in carry the sign that matches what the snowball is doing?
Which quantities are getting bigger, and which smaller? A rate carries that sign.
Is positive or negative for a melting snowball?
Theory UBC Math 100 §5CLP-1 §3.2Key results
Corrected working
Strategy
Differentiate the volume relation with respect to time, then substitute the instant’s values with their signs.
Working
Chain rule.
The radius decreases.
The volume is falling at about 226 cm³ per minute.
Answer
The answer is negative 72 pi cubic centimetres per minute, about negative 226.
Check
A melting snowball loses volume, so dV/dt must be negative; in 0.1 min r drops to 5.95 cm and V falls by about 22.4 cm³.
Where this goes wrong
Using +0.5 for a radius that is shrinking. A decreasing quantity has a negative rate; write the sign in before substituting.
Leans on
Volume of a sphere, from Functions, graphs and inverses
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