- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
The definition of the derivative practice question 16
A ball is thrown straight up from a balcony. Its height in metres after seconds is . Find the maximum height the ball reaches.
Give the answer in metres.
At the top of its flight the ball is momentarily at rest: its velocity, , is zero there.
For a quadratic the definition gives . Solve for , then evaluate at that time.
A decimal.
Theory UBC Math 100 §3.3CLP-1 §3.1Key results
Worked solution
Strategy
Velocity is the derivative of position. The ball rises while velocity is positive and falls once it is negative, so the maximum height is reached at the instant velocity is zero.
Working
Difference quotient of the quadratic; the constant 1.2 contributes nothing.
Velocity is zero one and a half seconds after the throw.
Height at that instant.
Answer
The answer is 12.225 metres.
Check
h(1.4) = h(1.6) = 12.176 m, equal and just below 12.225 m, so the peak sits halfway between them, at t = 1.5.
Where this goes wrong
Reporting the time 1.5 s, or forgetting the 1.2 m the ball started with. The question asks for a height, measured from the ground, not from the balcony.
Leans on
Evaluating a quadratic, from Functions, graphs and inverses
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