Algebra: Age Word Problems

Age problems are algebra word problems that deal with the ages of people currently, in the past or in the future.

Example 1:

Five years ago, John’s age was half of the age he will be in 8 years. How old is he now?

Solution:

Step 1: Let x be John’s age now. Look at the question and put the relevant expressions above it.

Step 2: Write out the equation.

Isolate variable x

Answer: John is now 18 years old.

Example 2:

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Solution:
Step 1: Set up a table.

	age now	age in 5 yrs
John
Peter
Alice

Step 2: Fill in the table with information given in the question.

John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three times as old as Alice. How old is Peter now?

Let x be Peter’s age now. Add 5 to get the ages in 5 yrs.

	age now	age in 5 yrs
John	2x	2x + 5
Peter	x	x + 5
Alice	x – 5	x – 5 + 5

Write the new relationship in an equation using the ages in 5 yrs.

In 5 years, John will be three times as old as Alice.

2x + 5 = 3(x – 5 + 5)

2x + 5 = 3x

Isolate variable x

x = 5

Answer: Peter is now 5 years old.

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Trigonometric Problems

In these lessons, we will learn the trigonometric functions (sine, cosine, tangent) and how to solve word problems using trigonometry.

Hints on solving trigonometry problems:

• If no diagram is given, draw one yourself.
• Mark the right angles in the diagram.
• Show the sizes of the other angles and the lengths of any lines that are known
• Mark the angles or sides you have to calculate.
• Consider whether you need to create right triangles by drawing extra lines. For example, divide an isosceles triangle into two congruent right triangles.
• Decide whether you will need the Pythagorean theorem, sine, cosine or tangent.
• Check that your answer is reasonable. The hypotenuse is the longest side in a right triangle.

Cosine

Example:  

Calculate the value of cos θ in the following triangle.

Solution:

Use the Pythagorean theorem to evaluate the length of PR.

Tangent

Example: 

Calculate the length of the side x, given that tan θ = 0.4

Solution:

Sine

Example: 

Calculate the length of the side x, given that sin θ = 0.6

Solution:

Using the Pythagorean theorem:

Solve Word Problems using Trigonometry

The following video shows how to use the trigonometric ratio, tangent, to find the height of a balloon.

This video shows how to use the trigonometric ratio, sine, to find the elevation gain of a hiker going up a slope.
A hiker is hiking up a 12 degrees slope. If he hikes at a constant rate of 3 mph, how much altitude does he gain in 5 hours of hiking?

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Trigonometry: Height of a tower

David measured the angle of elevation of the top of the tower from a point A to be 42◦ . He then moved 30 m closer to the tower and from point B measured the angle of elevation to the top of the tower to be 50◦ . To the nearest metre, determine the height of the tower. 

ABTˆ and T BCˆ form a straight angle. Therefore ABTˆ = 180◦ − 50◦ = 130◦ .
The angles in a triangle add to 180◦ so in 4T BA, AT Bˆ = 180◦ − 42◦ − 130◦ = 8◦ .
Let x represent the length of side BT and h represent T C, the required height.

Using the Sine Rule in 4ABT, x/sin 42◦ = 30/sin 8◦ and x = 30 sin 42◦/sin 8◦ ˙= 144.24.
Then in 4TBC, h/x = sin 50◦ and h = x sin 50 ˙= 110

The height of the tower is 110 m.

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Trigonometry: A 12.4 m flagpole is placed on top of a tall building

A 12.4 m flagpole is placed on top of a tall building. An observer, standing directly in front of the building and flagpole, measures the angle of elevation to the bottom of the flagpole to be 42.5 ◦ and to the top of the flagpole to be 48.2 ◦ . Determine the height of the building, to the nearest meter.

Solution:
Represent the given information on a diagram. Let the height of the building be h and the distance out from the building to the observer be d.

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Algebra

Algebra  is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols and is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are calledelementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

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