A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time $t$ seconds after she leaps, the horizontal distance from the front edge of the platform to the diver's shoulders is given by $x(t)$, and the vertical distance from the water surface to her shoulders is given by $y(t)$, where $x(t)$ and $y(t)$ are measured in meters. Suppose that the diver's shoulders are 11.4 meters above the water when she makes her leap and that $$\frac{dx}{dt} = 0.8 \quad \text{and} \quad \frac{dy}{dt} = 3.6 - 9.8t,$$ for $0 \leq t \leq A$, where $A$ is the time that the diver's shoulders enter the water.
(a) Find the maximum vertical distance from the water surface to the diver's shoulders.
(b) Find $A$, the time that the diver's shoulders enter the water.
(c) Find the total distance traveled by the diver's shoulders from the time she leaps from the platform until the time her shoulders enter the water.
(d) Find the angle $\theta$, $0 < \theta < \frac{\pi}{2}$, between the path of the diver and the water at the instant the diver's shoulders enter the water.

32. A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of 300 . After some time, the angle of depression becomes 600. The distance (in metres) travelled by the car during this time is :
(A) $100 \sqrt { 3 }$
(B) $( 200 \sqrt { } 3 ) / 3$
(C) $( 100 \sqrt { } 3 ) / 3$
(D) $200 \sqrt { } 3$

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If after five seconds, its elevation from the same point is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 720
(2) 1500
(3) 750
(4) 1440

An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt { 3 } \mathrm {~km}$ above it, is observed at an elevation of $60 ^ { \circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is $30 ^ { \circ }$, then the speed (in $\mathrm { km } / \mathrm { hr }$ ) of the aeroplane is
(1) 1500
(2) 750
(3) 720
(4) 1440

5. A red flag and a white flag are placed in a plaza, and Xiaoming stands between them. Using instruments on hand, Xiaoming measures that his distance to the red flag due east is 6 times his distance to the white flag due west. After walking 10 meters due north and measuring again, he finds that his distance to the red flag is now 4 times his distance to the white flag. What is the distance between the red and white flags closest to?
(1) 60 meters
(2) 65 meters
(3) 70 meters
(4) 75 meters
(5) 80 meters

II. Multiple-Choice Questions (35 points)

Instructions: For questions 6 through 12, each of the five options is independent, and at least one option is correct. Select all correct options and mark them on the ``Answer Sheet''. No deductions are made for incorrect answers. Five points are awarded for all five options correct; 2.5 points for one incorrect option; no points for two or more incorrect options.