IB Mathematics SL 2.5 Linear and other models AI HL Paper 1- Exam Style Questions- New Syllabus
Question
The trajectory of a dolphin leaping from the water is modelled by a quadratic function. The variables are defined as:
- \( d \): horizontal distance (in metres) from the launch point.
- \( h \): height (in metres) of the dolphin above the water surface.
Most-appropriate topic codes (IB Mathematics: Applications and Interpretation HL 2025):
• SL 2.2: Concept of a function, domain, range and graph – the concept of a function as a mathematical model — part (d)
▶️ Answer/Explanation
(a)
The axis of symmetry lies midway between the two points with the same \( h \)-value.
Midpoint = \( \frac{3 + 8.5}{2} = 5.75 \).
Equation of axis: \( d = 5.75 \).
\( \boxed{d = 5.75} \)
(b)
Using the axis formula for \( h(d) = -0.2d^2 + bd \):
Axis: \( d = -\frac{b}{2(-0.2)} = \frac{b}{0.4} = 2.5b \).
Set equal to 5.75: \( 2.5b = 5.75 \Rightarrow b = 2.3 \).
\( \boxed{2.3} \)
(c)
Re‑entry occurs when \( h(d) = 0 \):
\( -0.2d^2 + 2.3d = 0 \Rightarrow d(-0.2d + 2.3) = 0 \).
Non‑zero solution: \( d = \frac{2.3}{0.2} = 11.5 \).
\( \boxed{11.5 \, \text{m}} \)
(d)
A negative \( h \)-value would mean the dolphin is below the water surface. However, the model is defined only for the jump above water, so negative \( h \) is not physically meaningful within the domain of the model.
\( \boxed{\text{It would indicate the dolphin is underwater, which is outside the valid domain of the model.}} \)