January 2013

Area, moment of inertia, and radius of gyration of parabolic section

Situation
Given the parabola 3x² + 40y – 4800 = 0.

Part 1: What is the area bounded by the parabola and the X-axis?
A. 6 200 unit²
B. 8 300 unit²
C. 5 600 unit²
D. 6 400 unit²

Part 2: What is the moment of inertia, about the X-axis, of the area bounded by the parabola and the X-axis?
A. 15 045 000 unit⁴
B. 18 362 000 unit⁴
C. 11 100 000 unit⁴
D. 21 065 000 unit⁴

Part 3: What is the radius of gyration, about the X-axis, of the area bounded by the parabola and the X-axis?
A. 57.4 units
B. 63.5 units
C. 47.5 units
D. 75.6 units

Cylinder of maximum volume and maximum lateral area inscribed in a cone

Situation
A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius 6 m and height 12 m.

Part 1: Determine the radius of the cylinder such that its volume is a maximum.
A. 2 m
B. 4 m
C. 3 m
D. 5 m

Part 2: Determine the maximum volume of the cylinder.
A. 145.72 m³
B. 321.12 m³
C. 225.31 m³
D. 201.06 m³

Part 3: Determine the height of the cylinder such that its lateral area is a maximum.
A. 10 m
B. 8 m
C. 6 m
D. 4 m

Largest parabolic section from right circular cone

Situation
A right circular cone has a base diameter of 24 cm. The maximum area of parabolic segment that can be cut from this cone is 207.8 cm².

Part 1: Determine the base width of the parabola.
A. 22.32 cm
B. 18.54 cm
C. 15.63 cm
D. 20.78 cm

Part 2: Determine the altitude of the parabola.
A. 14 cm
B. 18 cm
C. 15 cm
D. 16 cm

Part 2: Determine the altitude of the cone.
A. 20 cm
B. 14 cm
C. 16 cm
D. 18 cm

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Depth of water in conical tank in upright and inverted positions

Situation
A closed conical vessel has a base radius of 2 m and is 6 m high. When in upright position, the depth of water in the vessel is 3 m.

Part 1: What is the volume of water?
A. 22 m³
B. 25 m³
C. 28 m³
D. 32 m³

Part 2: If the vessel is held in inverted position, how deep is the water?
A. 4.53 m
B. 5.74 m
C. 4 m
D. 5 m

Part 3: What is the weight of water in quintals. Unit weight of water is 9,800 N/m³.
A. 263.4
B. 195.4
C. 219.7
D. 247.2

10 Swimming pool in the shape of two intersecting circles

Situation
A swimming pool is shaped from two intersecting circles 9 m in radius with their centers 9 m apart.

Part 1: What is the area common to the two circles?
A. 85.2 m²
B. 63.7 m²
C. 128.7 m²
D. 99.5 m²

Part 2: What is the total water surface area?
A. 409.4 m²
B. 524.3 m²
C. 387.3 m²
D. 427.5 m²

Part 3: What is the perimeter of the pool, in meters?
A. 63.5 m
B. 75.4 m
C. 82.4 m
D. 96.3 m

02 Trapezoidal lot segregated from triangular land

Situation
A triangular lot ABC have side BC = 400 m and angle B = 50°. The lot is to be segregated by a dividing line DE parallel to BC and 150 m long. The area of segment BCDE is 50,977.4 m².

Part 1: Calculate the area of lot ABC.
A. 62,365 m²
B. 59,319 m²
C. 57,254 m²
D. 76.325 m²

Part 2: Calculate the area of lot ADE.
A. 8,342 m²
B. 14,475 m²
C. 6,569 m²
D. 11,546 m²

Part 3: Calculate the value of angle C
A. 57°
B. 42°
C. 63°
D. 68°

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Area of a spherical triangle with given interior angles

Problem
Find the area of a spherical triangle of whose angles are 123°, 84°, and 73°. The radius of the sphere is 30 m.

A. 1863.3 square meter
B. 1570.8 square meter
C. 1958.6 square meter
D. 1480.2 square meter

Geometric progression with some given terms

Situation
The 4^th term of a geometric progression is 6 and the 10^th term is 384.

Part 1: What is the common ratio of the G.P.?
A. 1.5
B. 3
C. 2.5
D. 2

Part 2: What is the first term?
A. 0.75
B. 1.5
C. 3
D. 0.5

Part 3: What is the seventh term?
A. 24
B. 32
C. 48
D. 96

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Calculator for Engineering Board Exam

Submitted by Jhun Vert on January 23, 2013 - 12:05pm

Any one who took an engineering board examination will agree with me that scientific calculator is our best friend during the examination. It will speed up our work and will allow us to go through the analysis of the problem. In most cases, solving the equation is not the real issue, it is to write the correct equation that matters. If we are able to determine the correct equation of a particular situation, the burden of that problem is over. Our next step is obviously to solve for the desired value. It is important to note that the correct answer can only be derived from the correct equation. To determine the correct equation is where should our brain go, not in finding the solution of that equation. You guess it right, we use our best friend, the scientific calculator, to do the task. Don't do manual calculation please, the examiner will not give you bonus points in finding the roots of

$9\sin^{3/4} \theta + 4\sin^{3/2} \theta - 12 = 0$

manually. Calculator can easily solve for the value of θ fast and accurate than you.

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Calculator Techniques for Solving Progression Problems

Submitted by Jhun Vert on January 27, 2013 - 11:22pm

This is the first round for series of posts about optimizing the use of calculator in solving math problems. The calculator techniques I am presenting here has been known to many students who are about to take the engineering board exam. Using it will save you plenty of time and use that time in analyzing more complex problems. The following models of CASIO calculator may work with these methods: fx-570ES, fx-570ES Plus, fx-115ES, fx-115ES Plus, fx-991ES, and fx-991ES Plus.