Functions and Angle

Q1.
Let hypotenus = h, perimeter of triangle = p and the other 2 sides of the right-angled triangle be a and b.
By Pythagoras theorem, a^2 + b^2 = h^2..................(1)
p = a + b + h..................(2)
By similar triangle, 12/a = b/h, that is 12h = ab............(3)
From (2),
(p - h)^2 = (a + b)^2 = a^2 + b^2 + 2ab
= h^2 + 24h.
p^2 + h^2 - 2ph = h^2 + 24h
p^2 - 2ph = 24h
p^2 = 24h + 2ph = h( 24 + 2p). Therefore,
h = p^2/(24 + 2p).
Q2.
When the minute hand moves 360 degree, the hour hand moves 30 degree.
For 4: 40, minute hand is at 240 degree from '12'.
Hour hand is [120 + (240/360)(30)] = 120 + 20 = 140 degree from '12'.
So smaller angle between the 2 hands = 240 - 140 = 100 degree = 5pi/9 radians.