Mathematics, Surveying and Transportation Engineering

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?

Problem
A parabola has an equation of y² = 8x. Find the equation of the diameter of the parabola, which bisect chords parallel to the line x – y = 4.

Problem
Given the Fourier equation:
 

$$f(t) = 5 \cos (20\pi t) + 2 \cos (40\pi t + \cos (80\pi t)$$
 

What is the fundamental frequency?

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.

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.

Problem
A point moves in the plane according to equations x = t² + 2t and y = 2t³ - 6t. Find dy/dx when t = 0, 2, 5.

Problem
A 523.6 cm³ 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.

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

Problem
Evaluate $\displaystyle \int_0^9 \dfrac{1}{\sqrt{1 + \sqrt{x}}}$