# Mathematics, Surveying and Transportation Engineering

Algebra, Trigonometry, Statistics, Geometry, Calculus, Differential Equations, Engineering Mechanics, Engineering Economy, Surveying, Transportation Engineering

## Rate of Change of Volume of Sand in Conical Shape

**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.

A. 2π ft/min | C. 3π ft/min |

B. 4π ft/min | D. 5π ft/min |

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## Duel of Two 50% Marksmen: Odds in favor of the man who shoots first

**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?

A. 1/3 | C. 2/3 |

B. 1/2 | D. 1/4 |

## Velocity of Separation: How fast is the distance between two cars changing?

**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.

A. 70 kph | C. 55 kph |

B. 80 kph | D. 60 kph |

## Centripetal Force of a Ball Revolving Uniformly in a Horizontal Circle

**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?

A. 74.95 m/sec^{2} |
C. 49.57 m/sec^{2} |

B. 94.75 m/sec^{2} |
D. 59.47 m/sec^{2} |

## Radius of Circle of New Atom Smasher

**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.

A. 1437 ft. | C. 1347 ft. |

B. 1734 ft. | D. 1374 ft. |

## Ratio of Volume of Water to Volume of Conical Tank

**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.

A. 4/5 | C. 26/27 |

B. 18/19 | D. 2/3 |

## Finding The Length Of Parabolic Curve Given Change In Grade Per Station

**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.

A. 300 m | C. 80 m |

B. 240 m | D. 120 m |

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

**Problem**

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

A. 2 | C. 4 |

B. -2 | D. -4 |

## Probability That A Randomly Selected Chord Exceeds The Length Of The Radius Of Circle

**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?

- 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 - 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 - 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

## Regular Octagon Made By Cutting Equal Triangles Out From The Corners Of A Square

**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?

A. 6.627 cm | C. 6.762 cm |

B. 6.267 cm | D. 6.276 cm |