MATH 1090 Algebra and Trigonometry
Problem Set 3 - Solutions

2002 Winter

Note

Determine whether the following equations are an identity, a conditional equation or an inconsistent equation and find the solution (where possible).
1. Clear denominators. LCD = 2 × 3 × 5 = 30
3. LCD = 6 Þ
4. The denominator x - 4 is zero when x = 4 . Exclude this value.
  
  But this value is excluded.

Today Abraham is three times the age of Zoe. In 20 years’ time, Abraham is twice the age of Zoe. How old are Abraham and Zoe today?

Solve the quadratic equations
1. Factorization:
  
  OR
  
  Quadratic Formula:
  $x = (-b +- sqrt{b^2 - 4ac}) / (2a)$
  
  OR
  
  Completing the Square:
2. Factorization:
  
  OR
  
  Quadratic Formula:
3. Factorization FAILS.
  Quadratic Formula:
  
  [There is a pair of complex conjugate solutions,

Solve
Check your proposed solution.

Square both sides

Square both sides again

Þ x = 2 OR x = 6

Check :

Solve
Check your proposed solution.

Let y² = x

Þ y = -3 OR y = 1

OR

Solve

Graph : (not required by the question)

Challenge Question

Elementary particles known as mu-mesons are created when cosmic rays hit the top of our atmosphere. They decay at rest in the time it takes light to travel only th of the way down to sea level. Yet these particles are observed at sea level.

Use Einstein’s time dilation formula

$T = To / sqrt{1 - v^2/c^2}$
to find the speed v, as a percentage of the speed of light c, at which the mu-mesons must travel in order for their decay time as measured by us to be dilated by a factor of 7.

Therefore the particles must travel at nearly of the speed of light.

Solve each of these inequalities and graph the solution set on a number line.

A projectile is fired from the ground into the air with an initial upward speed of 100 ms^-1. Assume that air resistance is negligible. How long will the projectile be in the air before it returns to the ground?

Note: to a good approximation, the vertical displacement y travelled upwards is related to the time elapsed t by

y = 100 t - 5 t²

Set y = 0 Þ 100 t - 5 t² = 0
Þ 5 t (20 - t) = 0
Þ t = 0 or t = 20.
t = 0 is when the projectile is fired.
t = 20 is when the projectile returns to the ground.
Therefore the projectile is airborne for 20 seconds.

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Dr. G.H. George