IB Mathematics SL 2.10 Solving equations AA SL Paper 2- Exam Style Questions- New Syllabus
Question
Two competitors, Diana and Persie, take part in a 200-meter dash on a linear track.
Diana’s velocity, \( v \), in \( \text{ms}^{-1} \), during the race is modeled by the function
\( \displaystyle v(t) = \frac{8.14t}{\sqrt{t^2 + 0.2}} \),
for \( t \geq 0 \), where \( t \) is the time in seconds from the start of the race.
(a) (i) State the value of \( v(1) \).
(ii) Determine the time when Diana reaches a velocity of \( 5 \, \text{ms}^{-1} \).
(b) Calculate the value of \( t \) at the instant Diana’s acceleration is \( 4 \, \text{ms}^{-2} \).
(c) (i) Find the limit of \( v(t) \) as \( t \) tends to infinity.
(ii) Provide a reason why this limiting value is not appropriate within the context of the race.
Persie’s velocity, \( w \), in \( \text{ms}^{-1} \), is modeled by the function
\( \displaystyle w(t) = \frac{8t}{\sqrt{t^2 + 0.3}} \),
for \( t \geq 0 \).
Diana wins the race. At the moment Diana reaches the 200-meter mark,
(d) Determine the distance Persie has remaining to reach the finish line.
Syllabus Component Reference:
• SL 2.10: Solving equations using technology or analytical methods — parts (a)(ii), (b)
• SL 5.1: Introduction to the concept of a limit — part (c)(i)
• SL 5.9: Kinematic problems involving displacement, velocity, and acceleration — part (d)
▶️ Answer/Explanation
(a)(i)
\( v(1) = \dfrac{8.14 \times 1}{\sqrt{1^2 + 0.2}} = \dfrac{8.14}{\sqrt{1.2}} \approx 7.43076 \)
\( \boxed{7.43 \, \text{ms}^{-1}} \)
(a)(ii)
Solve \( v(t) = 5 \):
\( \dfrac{8.14t}{\sqrt{t^2 + 0.2}} = 5 \)
\( 8.14t = 5\sqrt{t^2 + 0.2} \)
Square both sides: \( (8.14t)^2 = 25(t^2 + 0.2) \)
\( 66.2596t^2 = 25t^2 + 5 \)
\( 41.2596t^2 = 5 \)
\( t^2 \approx 0.1212 \)
\( t \approx 0.348 \) (seconds)
\( \boxed{0.348 \, \text{s}} \)
(b)
Acceleration \( a(t) = v'(t) \)
Using quotient rule or rewriting:
\( v(t) = 8.14t(t^2 + 0.2)^{-1/2} \)
\( v'(t) = 8.14(t^2 + 0.2)^{-1/2} – 8.14t^2(t^2 + 0.2)^{-3/2} \)
Set \( v'(t) = 4 \):
Solve numerically: \( t \approx 0.591 \) seconds
\( \boxed{0.591 \, \text{s}} \)
(c)(i)
As \( t \to \infty \), \( \sqrt{t^2 + 0.2} \to t \)
\( \displaystyle \lim_{t \to \infty} v(t) = \lim_{t \to \infty} \frac{8.14t}{t} = 8.14 \)
\( \boxed{8.14 \, \text{ms}^{-1}} \)
(c)(ii)
The race is finite (200 m) so Diana will stop running before infinite time.
OR
Diana cannot maintain that speed indefinitely in a real race.
Valid reason in context
(d)
Diana’s distance: \( \displaystyle s_F(t) = \int_0^t v(u) \, du \)
Integrate: \( \displaystyle \int \frac{8.14u}{\sqrt{u^2 + 0.2}} \, du = 8.14\sqrt{u^2 + 0.2} + C \)
Set \( s_F(t_F) = 200 \):
\( 8.14\sqrt{t_F^2 + 0.2} – 8.14\sqrt{0.2} = 200 \)
Solve: \( t_F \approx 25.0132 \) s
Persie’s distance in that time:
\( \displaystyle s_L(t_F) = \int_0^{t_F} w(t) \, dt = \int_0^{25.0132} \frac{8t}{\sqrt{t^2 + 0.3}} \, dt \)
\( = 8\sqrt{t^2 + 0.3} \Big|_0^{25.0132} = 8\sqrt{25.0132^2 + 0.3} – 8\sqrt{0.3} \approx 195.772 \) m
Distance from finish: \( 200 – 195.772 \approx 4.228 \) m
\( \boxed{4.23 \, \text{m}} \)