Mathematics, Surveying and Transportation Engineering

Problem
A conveyor is dispersing sands which forms into a conical pile whose height is approximately 4/3 of its base radius. Determine how fast the volume of the conical sand is changing when the radius of the base is 3 feet, if the rate of change of the radius is 3 inches per minute.

Problem
Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

Problem
A Toyota Land Cruiser drives east from point A at 30 kph. Another car, Ford Expedition, starting from B at the same time, drives S30°W toward A at 60 kph. B is 30 km from A. How fast in kph is the distance between two cars changing after 30 minutes? Hint: Use the Cosine Law.

Problem
A 150 g ball at the end of a string is revolving uniformly in a horizontal circle of radius 0.600 m. The ball makes 2 revolutions in a second. What is the centripetal acceleration?

Problem
A new kind of atom smasher is to be composed of two tangents and a circular arc which is concave toward the point of intersection of the two tangents. Each tangent and the arc of the circle is 1 mile long, what is the radius of the circle? Use 1 mile = 5280 ft.

Problem
A conical tank in upright position (vertex uppermost) stored water of depth 2/3 that of the depth of the tank. Calculate the ratio of the volume of water to that of the tank.

Problem
A +0.8% grade meets a -0.4% grade at km 12 + 850 with elevation 35 m. The maximum allowable change in grade per station is 0.2%. Determine the length of the curve.

Problem
Find y’ if x = 2 arccos 2t and y = 4 arcsin 2t.

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?