January 2012

Problem 01 | Right Spherical Triangle

Problem
Solve for the spherical triangle whose parts are a = 73°, b = 62°, and C = 90°.
 

Problem 03 | Inverse Laplace Transform

Problem 03
Find the inverse transform of   $\dfrac{7}{s^2 + 6}$.
 

Problem 04 | Inverse Laplace Transform

Problem 04
Perform the indicated operation:   $\mathcal{L}^{-1} \left[ \dfrac{s - 5}{s^2 + s - 6} \right]$
 

Problem 05 | Inverse Laplace Transform

Problem 05
Find the inverse transform of   $\dfrac{2s^2 + 5s - 6}{s^3 - 3s^2 - 13s + 15}$

Verbal Problems in Algebra

The following is an attempt to classify the verbal problems.
 

Number-related problems
Number-related problems are considered as the most basic type of verbal problems. It is taken as the base point of analysis for more complex type of problems.
 

Digit-related problems

Number-related Problems

Addition
Expressions that can be translated to addition, ( + ): sum, plus, added to, in addition, increased by, and more than.

Verbal expression Algebraic equivalent
the sum of x and y x + y or y + x
x plus y x + y
x increased by y x + y
x added to y y + x
x in addition to y y + x
x more than y y + x

 

Mixture-related Problems

There are four common types of mixture in verbal problems of Algebra.
 

Solutions
Solution is a homogeneous mixture formed by dissolving a substance (solute) in another substance (solvent). A common example is the salt as solute and water as solvent forming into one phase called brine or saline water.
 

Alloys
An alloy is a solid solution formed by fusing two or more metallic elements. A common alloy is bronze which is the product of fusing iron and copper.
 

Digit-Related Problems

For any three digit number, let
h = the hundreds digit
t = the tens digit and
u = the units digit
 

The number = 100h + 10t + u
The number with digits reversed = 100u + 10t + h
The sum of digits = h + t + u
The product of digits = htu
 

Motion-related Problems

Motion with constant velocity
The distance traveled is the product of velocity and time.
 

$s = vt$

were,
s = distance
v = velocity
t = time
 

It follows that

$t = \dfrac{s}{v}$   and   $v = \dfrac{s}{t}$

 

Age-related Problems

If x = present age of a person
x – 3 = age of the person 3 years ago
x + 5 = age of the person 5 years from now or 5 years hence
 

Note:
The difference of the ages of two persons is constant at any time.
 

Work-related Problems

Case 1: Workers have different rates

Work rate × Time to finish the job = 1 job done

Work rate = (1 job done) / (Time to finish the job)

Time of doing the job = (1 job done) / (Work rate)

 

For example
Albert can finish a job in A days
Bryan can finish the same job in B days
Carlo can undo the job in C days
 

Clock-related Problems

There are 12 dial units in the clock. Every time the minute hand completes 12 dials, the hour hand moves 1 dial. Thus, if the minute hand moves by x the hour hand moves by x/12.
 

000-clock-problem-illustration.gif

 

Key equations:

$x$ = distance traveled by the minute hand (in minutes)

$\dfrac{x}{12}$ = distance traveled by the hour hand (in minutes)

 

Money-related Problems

Coin denominations in the US

Name Coin value Dollar Equivalent
Penny 1 ¢ \$0.01
Nickel 5 ¢ \$0.05
Dime 10 ¢ \$0.10
Quarter 25 ¢ \$0.25
Half-dollar 50 ¢ \$0.50

 

Time When The Angle Between The Hands Of The Clock Were Bisected By The 3 O'clock Mark

Civil Engineering Math Refresher Final Preboard:

Problem
What time between 2 and 3 o’clock will the angle between the hands of the clock be bisected by the line connecting the center of the clock and the 3 o’clock mark?

A. 2:18:27.6
B. 2:16:00.0
C. 2:17:56.3
D. 2:19:03.1