Algebra

How Long Would it Take to Fly From Earth to Jupiter?

Problem
Earth is approximately 93,000,000.00 miles from the sun, and the Jupiter is approximately 484,000,900.00 miles from the sun. How long would it take a spaceship traveling at 7,500.00 mph to fly from Earth to Jupiter?

A.   9.0 years C.   6.0 years
B.   5.0 years D.   3.0 years

 

Amount of Sales Needed to Receive the Desired Monthly Income

Problem
A salesperson earns \$600 per month plus a commission of 20% of sales. Find the minimum amount of sales needed to receive a total income of at least \$1500 per month.

A.   \$1500 C.   \$4500
B.   \$3500 D.   \$2500

 

Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If x, a, y is a geometric progression then 'a' is called geometric mean. If x, a, y form a harmonic progression then 'a' is called harmonic mean.
 

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula
 

$AM \times HM = GM^2$

 

Below is the derivation of this relationship.
 

Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

  • 3, 6, 12, 24, … is a geometric progression with r = 2
  • 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2

 

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP
Definition

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.
 

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4

Sum and Product of Roots

The quadratic formula
 

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

 

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that
 

$x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a}$   and   $x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a}$

 

where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. The sum of roots x1 + x2 and the product of roots x1·x2 are common to problems involving quadratic equation.
 

Derivation of Quadratic Formula

The roots of a quadratic equation ax2 + bx + c = 0 is given by the quadratic formula
 

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

 

The derivation of this formula can be outlined as follows:

  1.   Divide both sides of the equation ax2 + bx + c = 0 by a.
  2.   Transpose the quantity c/a to the right side of the equation.
  3.   Complete the square by adding b2 / 4a2 to both sides of the equation.
  4.   Factor the left side and combine the right side.
  5.   Extract the square-root of both sides of the equation.
  6.   Solve for x by transporting the quantity b / 2a to the right side of the equation.
  7.   Combine the right side of the equation to get the quadratic formula.

See the derivation below.
 

 
 
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