A few challenge problems

[joml88]

Here are a few challenge problems to solve

(I don't know what the difficulty of this board is so I would appreciate comments)

 

1. u and w are in degrees. tan(U)=1/2 and tan(w) = 2 find tan(u-w)

 

2. There are 20 switches in two columns each with 10 in them. Each switch is either on or off. If exactly 5 must be on in each column what is the number of distinct ways the switches can be set?

 

3. A metal plate of constant thickness is cut into a rt. triangle. The coordinates are (0,0), (2,0) and (2,1). The plate is balanced on a fulcrum on the side which connects (0,0) and (2,0). Find the x coordinate of the balance point of the fulcrum.

[Dave]

1. tan(A-B) = (tan A - tan B) / (1 + tan A tan B)

 

So tan(u-w) = (tan(u) - tan(w))/(1+tan(u)tan(w)) = -3/2 / 2 = -3/4.

 

2. Number of combinations for one column = nCr(10, 5) => total number of ways the switches can be set = 2*nCr(10, 5). Can't be bothered working out nCr(10, 5) though

 

3. Just work out the co-ordinates of the centre of mass of the triangle (assuming its density is constant).

[paganinio]

how can a Rt.Triangle be balanced on a fulcrum on the side?

[paganinio]

joml88, thankyou for giving us such a good weblink, there i found the best math message board on the internet(what? that's not the best? tell me which is better!)

[BrainMan]

For 2, the number of ways to have one column of switches needs to be multiplied by the number of ways to have the other column of switches, which makes it [nCr(10, 5)]^2 rather than 2*nCr(10, 5)

 

For 3, this is a matter of finding where the area is 1/2 the origional area after cutting the triangle in two with a line of y = c. That is the same as taking the integral of y = x/2 from 0 to 2 (for the original area, A = 1) and then finding the integral of x/2 from 0 to u, setting it equal to 1/2 (half the original area) and solving for u. The answer is Sqrt(2).