Mathematics, Surveying and Transportation Engineering

Situation
If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

1. Assume that the distance of the chord from the center of the circle is uniformly distributed.

   A.   0.5	C.   0.866
   B.   0.667	D.   0.75

2. Assume that the midpoint of the chord is evenly distributed over the circle.

   A.   0.5	C.   0.866
   B.   0.667	D.   0.75

3. Assume that the end points of the chord are uniformly distributed over the circumference of the circle.

   A.   0.5	C.   0.866
   B.   0.667	D.   0.75

Problem
A regular octagon is made by cutting equal isosceles right triangles out from the corners of a square of sides 16 cm. What is the length of the sides of the octagon?

Problem
How many minutes after 3:00 o’clock will the hands of the clock be perpendicular to each other for the 1st time?

Problem
Divide the circle of radius 13 cm into four parts by two perpendicular chords, both 5 cm from the center. What is the area of the smallest part.
 

Problem
A certain businessman, who is always in a hurry, walks up an ongoing escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

Problem
Player M has Php1, and Player N has Php2. Each play gives one the players Php1 from the other. Player M is enough better than player N that he wins 2/3 of the plays. They play until one is bankrupt. What is the chance that Player M wins?

Problem
What is the angle between zero-based vectors ${\bf V_1} = (-\sqrt{3}, ~ 1)$ and ${\bf V_2} = (2\sqrt{3}, ~ 2)$ in an x-y coordinate system?

Problem
In a common carnival game, a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

Problem
ABCD is a square of side 10 cm. PQRS is a square inside ABCD. PQBA, QRCB, RSDC, and SPAD are identical trapezia, each of area 16 cm². What is the height of each trapezium if PQ is parallel to AB and SR is parallel to DC?

Problem
From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?