MATHalino

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
 

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°
 

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
 

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

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
 

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
 

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
 

Summary for the value of end moments and deflection of perfectly restrained beam carrying various loadings. Note that for values of EIy, y is positive downward.
 

Case 1: Concentrated load anywhere on the span of fully restrained beam

End moments
$M_A = -\dfrac{Pab^2}{L^2}$

$M_B = -\dfrac{Pa^2b}{L^2}$
 

Value of EIy
$\text{Midspan } EI\,y = \dfrac{Pb^2}{48}(3L - 4b)$

Note: only for b > a

Problem 737
In the perfectly restrained beam shown in Fig. P-737, support B has settled a distance Δ below support A. Show that M_B = -M_A = 6EIΔ/L².