Mathematics, Surveying and Transportation Engineering

Problem
The area of a park on a map is 600 mm². If the scale of the map is 1 to 40,000 determine the true area of the park in hectares (1 hectare = 10⁴ m²).

Problem
Evaluate $\displaystyle \int_0^1 \dfrac{x \, dx}{x^2 + 2}$.

Problem
Determine the radius of curvature of the curve $x = y^3$ at point (1, 1).

Problem
Calculate the area enclosed by the curve $x^2 + y^2 - 10x + 4y - 196 = 0$.

Problem
The first three terms of a geometric progression are 2x, 4x + 14 and 20x - 14. Find the sum of the first ten terms.

Problem
A 50-m steel tape that is is 0.02 m too long was used to measure the distance between two points A and B. If the measured distance was 160.42 m, calculate the correct distance between A and B.

Problem
A circle has an equation of $x^2 + y^2 + 2cy = 0$. Find the value of $c$ when the length of the tangent from (5, 4) to the circle is equal to one.

Problem
Calculate the acute angle between two intersecting surfaces whose equations are as follows:
$$2x - 4y - z = -5$$

$$3x + 4y + 5z = -6$$

Problem
If $\arcsin (3x - 4y) = 1.571$ and $\arccos (x - y) = 1.047$, what is the value of $x$?

Problem
The digits of a three-digit number are in arithmetic progression. If you divide the number by the sum of its digits, the quotient is 26. If the digits are reversed, the resulting number is 198 more than the original number. Find the sum of all the digits.