Mathematics, Surveying and Transportation Engineering

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Four Trapezia Formed by the Difference of Two Concentric Squares

ABCD is a square of side 10 cm. PQRS is a square inside ABCD. PQBA, QRCB, RSDC, and SPAD are identical trapezia, each of area 16 cm2. What is the height of each trapezium if PQ is parallel to AB and SR is parallel to DC?

A.   3 cm C.   2 cm
B.   1.8 cm D.   1.2 cm




Random Steps of a Drunk Man: Probability of Escaping the Cliff

From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff 1/3. What is his chance of escaping the cliff?

A.   2/27 C.   4/27
B.   107/243 D.   1/2


Length of Parabolic Sag Curve with Given Change in Grade Per Station

A grade of -5% is followed by a grade of 1%, the grades intersecting at the vertex (Sta. 10 + 060). The change of grade is restricted to 0.4% in 20 m. Compute the length of the vertical parabolic sag curve in meters.

A.   360 m C.   300 m
B.   320 m D.   340 m


The Distance the Particle Travels with Given Position Function x(t) = t^4 - 8t^2

Given the position function x(t) = t4 - 8t2, find the distance that the particle travels at t = 0 to t = 4.

A.   160 C.   140
B.   150 D.   130


Answer Key


Compound Curves: Finding the Stationing of PCC with Given Stationing of PC

A compound curve has the following characteristics:

I1 = 24° D1 = 6°
I2 = 36° D2 = 4°
Stationing of P.C. = km 10 + 420

Compute the stationing of P.C.C.

A.   km 10 + 560 C.   km 10 + 520
B.   km 10 + 540 D.   km 10 + 500

For Sn = 3^(2n - 1) + b; Find the Quotient a9 / a7

The sum of the first n terms of a series is 3^(2n - 1) + b. What is the quotient of the 9th and the 7th term?

A.   81 C.   83
B.   82 D.   84


Y-component of Vector A if Vectors A and B are Orthogonal

Compute the value of b if A and B are orthogonal.
$${\bf A} = 2{\bf i} + b{\bf j} + {\bf k}$$

$${\bf B} = 4{\bf i} - 2{\bf j} - 2{\bf k}$$

A.   6 C.   4
B.   5 D.   3


General Term of Arithmetic Sequence that Models the Potential Annual Salaries

A job posted at offered a starting salary of \$40,000 per year and guaranteeing a raise of \$1600 per year for the rest of 5 years. Write the general term for the arithmetic sequence that models potential annual salaries.

A.   an = 38,400 + 1600n
B.   an = 33,400 + 2600n
C.   an = 36,400 + 1400n
D.   an = 34,400 +1800n

Weight of Copper and Tin to Produce an Alloy of 30% Copper and 10% of Tin

A given alloy contains 20% copper and 5% tin. How many pounds of copper and of tin must be melted with 100 lb of the given alloy to produce another alloy analyzing 30% copper and 10% tin? All percentages are by weight.

A.   20.5 lb copper and 4.5 lb tin
B.   17.5 lb copper and 7.5 lb tin
C.   19.5 lb copper and 5.5 lb tin
D.   18.5 lb copper and 6.5 lb tin




Nutrition: Amount of food A, B and C to Meet the Nutrient Requirements

A nutritionist in a hospital is arranging special diets that consist of a combination of three basic foods. It is important that the patients on this diet consume exactly 310 units of calcium, 190 units of iron, and 250 units of vitamin A each day. The amounts of these nutrients in one ounce food are given in the following table.

  Units Per Ounce
Calcium Iron Vitamin A
Food A 30 10 10
Food B 10 10 30
Food C 20 20 20


How many ounces each food must be used to satisfy the nutrient requirements exactly?

A.   6 ounces of Food A, 5 ounces of Food B and 3 ounces of Food C
B.   3 ounces of Food A, 5 ounces of Food B and 6 ounces of Food C
C.   6 ounces of Food A, 3 ounces of Food B and 5 ounces of Food C
D.   5 ounces of Food A, 3 ounces of Food B and 6 ounces of Food C



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