IB Mathematics SL 2.5 Linear models AI SL Paper 1- Exam Style Questions- New Syllabus
Question
In a baseball game, Keiko is the batter standing beside home plate. The ball is thrown towards home plate along a path that can be modelled by \[ y=-0.045x+2 . \] In this model, \(x\) is the horizontal distance of the ball from the point the ball is thrown and \(y\) is the vertical height of the ball above the ground (both in metres).
The outcome of the throw is called a strike if the height of the ball is between \(0.53\ \text{m}\) and \(1.24\ \text{m}\) at some point while it travels over home plate. The length of home plate is \(0.43\ \text{m}\).
When the ball reaches the front of home plate, the height of the ball above the ground is \(1.25\ \text{m}\). The height of the ball changes by \(a\) metres as the ball travels over the length of home plate.
(a) (i) Find the value of \(a\). [2]
(ii) Justify why this throw is a strike. [2]
On the next pitch, Keiko hits the ball towards a wall that is \(5\) metres high. The horizontal distance of the wall from the point where the ball was hit is \(96\) metres. The path of the ball after it is hit can be modelled by \[ h(d)=-0.01d^{2}+1.04d+0.66,\quad h,d>0, \] where \(h\) is the height of the ball above the ground and \(d\) is the horizontal distance of the ball from the point where it was hit (metres).
(b) Determine whether the ball will go over the wall. Justify your answer. [3]
▶️Answer/Explanation
Markscheme
(a)
(i) Change in height across the plate of length \(0.43\) m is \[ a=\text{slope}\times\Delta x=(-0.045)\times 0.43=-0.01935\ \text{m}\ (\approx -0.0194\ \text{m}). \] M1 A1
(ii) Back edge height \[ y_{\text{back}}=1.25+a=1.25-0.01935=1.23065\ \text{m}. \] Since \(0.53<1.23065<1.24\), the ball passes through the strike zone while over the plate, so it is a strike. A1 R1
[4 marks]
(b)
Evaluate height at the wall: \[ \begin{aligned} h(96)&=-0.01\times 96^{2}+1.04\times 96+0.66 \\ &=-0.01\times 9216+99.84+0.66 \\ &=-92.16+100.50=8.34\ \text{m}. \end{aligned} \] Since \(8.34>5\), the ball clears the wall. M1 A1 A1
[3 marks]
Total Marks: 7