Help with maths please >.<!

Here's a few questions that I can't get :mad: !

Q1 At 6pm the hands of a clock are 180o apart. How long will it be before the hands of the clock are again 180o... I know how to get the answer but I don't know how to prove me answer mathemetically.

Q2 A 2L can is an exact enlargement of a one litre can. Explain why their base radii are in the ratio of 1 : (Cube Root of 2).

Q3 If N+2 is divisible by 3 and 2N-1 is divisible by 7, prove that N+17 is divisble by 21.

Thanks for your help :wave: !

Q1) Let:
a the angle from the vertical line to the final position of either hand
t the time to be found
w₁ the angular velocity of the upper hand
w₂ the angular velocity of the lowerhand

Then a can be written as:

a = w₂t
and
a + 2*Pi = w₁t

where the angle is expressed in radians.

From this,

w₂t = w₁t - 2*Pi and solving for t:

t = 2*Pi / (w₁ - w₂)

The lower (slow) hand takes 6 hours to complete an angle of 180 deg. or Pi, so that:

w₂ = Pi/6 h^-1

whereas the upper hand takes 1 hour to complete one full of turn of 360 deg. (2*Pi). Thus:

w₁ = 2*Pi h^-1

Substituting,

t = 2*Pi / (2*Pi h^-1 - Pi/6 h^-1) h = 12/11 h = 1.0909... h = 1 hour, 5 min. 7.2727... sec.

Q2) An exact enlargement means all dimensions are enlarged by the same factor, i.e.

R = kr
H = kh

where R and r and the radii and H and h the heights and k is an enlargement factor.

The volumes are:

v = Pi*r²h
V = Pi*R²H = Pi*(kr)²(kh)²

Because V = 2v,

Pi*(kr)²(kh)² = 2Pi*r²h

Hence,

k³ = 2

Q3) Prove that (N + 17) is divisible by both 3 and 7.

(N + 17) - (N + 2) = 15; Divisible by 3
2(N + 17) - (2N - 1) = 35; Divisible by 7