Fun math questions!?

Question

six people are going to sit in a row on a bench. Romeo wants to sit next to Juliet. Caesar does not want tosit next to Brutus. Homer and Pierre can sit anywhere. How many ways can these six people be seated?

If the length of the hour and the minute hands of a clock are 4 cm and 6 cm, respectively, what is the distance in cm between the tips of the hands at two o'clock?

Can you answer these?

Answer 1

Actually, at 2:00 the hands are 60 degrees apart. The cosine of 60 degrees is 1/2. So the first answer should be modified accordingly.

As for the first question, I'd divide it into four cases

A. R&J at the end.
B. R&J one spot from the end, and one of C or B at the end next to them.
C. R&J one spot from the end, and both of C and B in the 3 spots on the other side of them.
D R&J in the middle.

For B there are 4 arrangements for R&J, 2 for the C or B next to them, the 3 for the other of C or B, and the 2 for H & J. So there are 4x2x3x2 versions of Case B total.

For C it's similarly 4 x 2 x 2.

For D it's 2 (for R & J) x 2 (for which of B and C is to the right of R&J and which is to the left) x 2 (two choices for where B sits) x 2 (two choices for where C sits) x 2 (two ways to arrange H&J)

And so on. At the end, add up all the possibilities.

Answer 2

Hello,

Second one.

Use the law of cosines.

c ^2 = a^2 + b^2 -2ab cos(angle between a and b)

At 20 o'clock the hands are 10 minutes apart or 60minutes / 10 = 6 degrees apart.

So we have 6^2 + 4^2 - 2*6*4 cos 6 = c^2

36 + 16 -48*0.9945 = c^2 giving us

4.264 = c^2 so c = 2.065

Hope This Helps!