Steven, who loves soccer, trains his shooting skills by trying to kick balls into the net from dista

Question

Steven, who loves soccer, trains his shooting skills by trying to kick balls into the net from distances varying from $15$ metres to $35$ metres. The probability, $p$, that he kicks the ball into the net is given by $p=\frac{2}{65}\left( 40-d \right)$, where $d$ is the distance between the ball and the goalpost in metres.
Each kick by him is assumed to be independent of previous kicks.[2]
1. Find the maximum value of $d$ (to the nearest integer) for there to be more than $60\%$ chance of Steven kicking the ball into the net.[2]
2. Steven kicks $15$ balls from a fixed distance of $x$ metres from the goalpost.
  Find, to the nearest integer, the value of $x$ such that Steven has $90\%$ chance of kicking at least two balls into the net.
  [4]

Accepted Answer

(b)(i) $15\le d<20.5$ m, with no largest real distance. Source-intended greatest whole-metre distance: $20$ m (ii) $x\approx32.3440$ m, giving $32$ m to the nearest integer.

2015 AJC P2 Q11

2015 AJC P2 Q11

Solution:

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