In a shooting game, the player has to shoot with a gun to capture a monster. The bullet from the gun has to land on a circular target on the monster so that a net will open up. The circular target has four different sizes. Target 1 is a big circle; Target 2 is three quarters of the area of Target 1; Target 3 is half of the area of Target 1; Target 4 is one quarter of the area of Target 1. Even if the bullet hits the circular target, the monster might not be captured as it can break free from the net. The probability of successfully capturing the monster depends on the area and colour of the circular target, and the type of bullet that the player uses. Information that the players need is given below.

1. (a)

   Sam estimates that the probability that he hits Target 1 is about 0.88. Given that the probability of hitting the circular target is proportional to its area, show that the probability that Sam hits Target 2 and Target 4 is estimated to be 0.66 and 0.44 respectively.

   Answer: ...

2. (b)

   Show that the estimated probability that Sam will hit and capture a monster with an orange Target 4 using a Flat Nose Bullet is 0.0495.

3. (c)

   At time $t = 0$ seconds, Sam sees a monster with a red Target 2. He only has one Soft Point Bullet. State the colour, Target size and find the maximum probability and the number of seconds Sam should take to shoot at the monster in order to maximise his chance of hitting and capturing it. Justify your decision and show your calculations clearly.