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Question 1
A ball is thrown so that its height above the ground is modelled by $h = - 4.9 t^{2} + 12 t + 1.5$ , where $h$ is in metres and $t$ is in seconds, valid until the ball lands. Given the ball hits the ground when $t \approx 2.57$ s, what is the most reasonable domain for this model?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $0 \leq t \leq 2.57$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Real-world constraint
Time cannot be negative, so the model starts at $t = 0$ , the moment the ball is thrown.
Step 2: End condition
The model stops being meaningful once the ball hits the ground, which occurs at $t \approx 2.57$ s (where $h = 0$ again).
Step 3: Combine bounds
The valid domain is therefore $0 \leq t \leq 2.57$ .
Step 4: Context check
Outside this range the height values would be physically meaningless (negative height or before the ball was thrown).
Method #2Why the others are wrong
Step 1: All real numbers
$t \in R$ ignores that negative time and time after landing are not physically meaningful.
Step 2: Negative lower bound
$- 2.57 \leq t \leq 2.57$ incorrectly allows negative time, which has no meaning before the throw.
Step 3: Confusing coefficient with bound
$0 \leq t \leq 12$ mistakenly uses the coefficient 12 from the equation rather than the actual landing time.
Step 4: Correct choice
Only $0 \leq t \leq 2.57$ reflects the physical situation correctly.
Question 2
A shop owner records weekly sales of umbrellas against average weekly rainfall. As rainfall increases, sales increase, and the increase between successive equal rainfall intervals is roughly the same each time (constant absolute increase, not percentage). Which model type is most appropriate?
No clue? Show me the answer
Correct answerCorrect!Incorrect
BLinear
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Key feature
The description states the increase is constant for equal rainfall intervals, meaning a constant rate of change.
Step 2: Match to model type
A constant rate of change is the defining feature of a linear model $y = m x + b$ .
Step 3: Check other features
There is no mention of percentage growth, a turning point, or periodicity, ruling out the other model types.
Step 4: Conclusion
The most appropriate model is linear.
Method #2Why the others are wrong
Step 1: Exponential mistake
Exponential requires constant percentage change, not constant absolute change as described.
Step 2: Quadratic mistake
Quadratic requires a single turning point (increase then decrease), which is not described here.
Step 3: Sinusoidal mistake
Sinusoidal requires a repeating, cyclical pattern, which does not apply to a steadily increasing trend.
Step 4: Correct choice
Linear is the only model matching a constant rate of change.

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Free preview 2/16

Question 1
A ball is thrown so that its height above the ground is modelled by $h = - 4.9 t^{2} + 12 t + 1.5$ , where $h$ is in metres and $t$ is in seconds, valid until the ball lands. Given the ball hits the ground when $t \approx 2.57$ s, what is the most reasonable domain for this model?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $0 \leq t \leq 2.57$
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Real-world constraint
Time cannot be negative, so the model starts at $t = 0$ , the moment the ball is thrown.
Step 2: End condition
The model stops being meaningful once the ball hits the ground, which occurs at $t \approx 2.57$ s (where $h = 0$ again).
Step 3: Combine bounds
The valid domain is therefore $0 \leq t \leq 2.57$ .
Step 4: Context check
Outside this range the height values would be physically meaningless (negative height or before the ball was thrown).
Method #2Why the others are wrong
Step 1: All real numbers
$t \in R$ ignores that negative time and time after landing are not physically meaningful.
Step 2: Negative lower bound
$- 2.57 \leq t \leq 2.57$ incorrectly allows negative time, which has no meaning before the throw.
Step 3: Confusing coefficient with bound
$0 \leq t \leq 12$ mistakenly uses the coefficient 12 from the equation rather than the actual landing time.
Step 4: Correct choice
Only $0 \leq t \leq 2.57$ reflects the physical situation correctly.
Question 2
A shop owner records weekly sales of umbrellas against average weekly rainfall. As rainfall increases, sales increase, and the increase between successive equal rainfall intervals is roughly the same each time (constant absolute increase, not percentage). Which model type is most appropriate?
No clue? Show me the answer
Correct answerCorrect!Incorrect
BLinear
Step-by-step walkthrough
Choose a solution method
Method #1Worked solution
Step 1: Key feature
The description states the increase is constant for equal rainfall intervals, meaning a constant rate of change.
Step 2: Match to model type
A constant rate of change is the defining feature of a linear model $y = m x + b$ .
Step 3: Check other features
There is no mention of percentage growth, a turning point, or periodicity, ruling out the other model types.
Step 4: Conclusion
The most appropriate model is linear.
Method #2Why the others are wrong
Step 1: Exponential mistake
Exponential requires constant percentage change, not constant absolute change as described.
Step 2: Quadratic mistake
Quadratic requires a single turning point (increase then decrease), which is not described here.
Step 3: Sinusoidal mistake
Sinusoidal requires a repeating, cyclical pattern, which does not apply to a steadily increasing trend.
Step 4: Correct choice
Linear is the only model matching a constant rate of change.

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SL 2.6—Modelling skills

Step-by-step walkthrough

Step 1: Real-world constraint

Step 2: End condition

Step 3: Combine bounds

Step 4: Context check

Step 1: All real numbers

Step 2: Negative lower bound

Step 3: Confusing coefficient with bound

Step 4: Correct choice

Step-by-step walkthrough

Step 1: Key feature

Step 2: Match to model type

Step 3: Check other features

Step 4: Conclusion

Step 1: Exponential mistake

Step 2: Quadratic mistake

Step 3: Sinusoidal mistake

Step 4: Correct choice

14 more questions in this topic

SL 2.6—Modelling skills

Step-by-step walkthrough

Step 1: Real-world constraint

Step 2: End condition

Step 3: Combine bounds

Step 4: Context check

Step 1: All real numbers

Step 2: Negative lower bound

Step 3: Confusing coefficient with bound

Step 4: Correct choice

Step-by-step walkthrough

Step 1: Key feature

Step 2: Match to model type

Step 3: Check other features

Step 4: Conclusion

Step 1: Exponential mistake

Step 2: Quadratic mistake

Step 3: Sinusoidal mistake

Step 4: Correct choice

14 more questions in this topic