# Mathematics, Surveying and Transportation Engineering

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

## Samuel Pepys Wrote Isaac Newton Asking Which Event is More Likely to Occur

**Problem**

Samuel Pepys wrote Isaac Newton to ask which of three events is more likely: that a person get (*a*) at least 1 six when 6 dice are rolled (*b*) at least two sixes when 12 dice are rolled, or (*c*) at least 3 sixes when 18 dice are rolled. What is the answer?

*a*) is more likely than (

*b*) and (

*c*)

B. (

*b*) is more likely than (

*a*) and (

*c*)

C. (

*c*) is more likely than (

*a*) and (

*b*)

D. (

*a*), (

*b*), and (

*c*) are equally likely

## Burning Bale from Catapult to Travel 150 ft and Clear Over 35 ft Castle Wall

**Problem**

A catapult is placed 100 ft from the castle wall, which is 35 feet high. The soldier wants the burning bale of hay to clear the top of the wall and land 50 feet inside the castle wall. If the initial velocity of the bale is 70 feet per second, then at what angle should the bale of hay be launched so that it travel 150 feet and pass over the castle wall. Use *g* = 32 ft/sec^{2}.

A. 49.8° | C. 39.2° |

B. 50.8° | D. 40.2° |

## Spinning Spherical Target: Probability for Three Marksmen to Hit on the Same Hemisphere

**Problem**

Three marksman simultaneously shoot and hit a rapidly spinning spherical target. What is the probability that the three points of impact lie on the same hemisphere?

A. 0 | C. 1 |

B. 1/2 | D. 2/3 |

## Equation of the Diameter of Parabola Bisecting Parallel Chords of Given Slope

**Problem**

A parabola has an equation of *y*^{2} = 8*x*. Find the equation of the diameter of the parabola, which bisect chords parallel to the line *x* – *y* = 4.

A. y = 2 |
C. y = 4 |

B. y = 3 |
D. y = 1 |

## Fundamental Frequency of Fourier Equation in Cosine Form

**Problem**

Given the Fourier equation:

*f*(

*t*) = 5 cos (20π

*t*) + 2 cos (40π

*t*) + cos (80π

*t*)

What is the fundamental frequency?

A. 10 | C. 40 |

B. 20 | D. 30 |

## Amount of Sales Needed to Receive a Specified Monthly Income

**Problem**

A salesperson earns P60,000 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total income of at least P150,000 per month.

A. P150,000 | C. P450,000 |

B. P350,000 | D. P250,000 |

## Area Bounded by Intersecting Chords in a Circle

**Problem**

Chords *AB* and *CD* intersect each other at *E* inside the circle. *AE* = 8 cm, *CE* = 12 cm, and *DE* = 20 cm. If *AB* is the diameter of the circle, compute the area of *AEC*.

A. 61.04 cm^{2} |
C. 39.84 cm^{2} |

B. 52.05 cm^{2} |
D. 48.62 cm^{2} |

## Slope of a Curve of Given Parametric Equations

**Problem**

A point moves in the plane according to equations *x* = *t*^{2} + 2*t* and *y* = 2*t*^{3} - 6*t*. Find *dy*/*dx* when *t* = 0, 2, 5.

A. -3, -3, -12 | C. 3, 3, 12 |

B. 3, -3, 12 | D. -3, 3, 12 |

## A solid spherical ball remolded into a hollow spherical ball

**Problem**

A 523.6 cm^{3} solid spherical steel ball was melted and remolded into a hollow steel ball so that the hollow diameter is equal to the diameter of the original steel ball. Find the thickness of the hollow steel ball.

A. 1.3 cm | C. 1.2 cm |

B. 1.5 cm | D. 1.6 cm |

## Distance From a Point to a Plane in 3D-Space

**Problem**

Find the distance from the point *A*(1, 5, -3) to the plane 4*x* + *y* + 8*z* + 33 = 0.

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

B. 2 | D. 1.5 |