College Mathematics
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Examples and Solutions

Interest: You plan to put away $50.00 every month in the bank at a rate of 8% compounded monthly. How much will you have at the end of 5 years?

This is a problem on Annuity. It is a sequence of equal deposits or payments toward an account made at equal time intervals. The Amount of the Annuity will be the total at the end of the period including all interests. The amount of an annuity compounded annually is given by:
                                                                 P{(1 + i)N - 1}
                                                       V =   --------------------
                                                                 i
where P in dollars is the invested amount at the end of each year up to N years at interest rate i. If the interest is compounded n times each year and deposits made every compounding period, then                          
                                                                         P{(1 + i/n)nN - 1}
                                                               
 V =  ------------------------                                                                        i/n

the formula to be used in this example. Substituting the values, we then have

                                              50{(1 + 0.08/12)(12X5) - 1}              50{(1 + 0.006667)60 - 1}
                                  V  =   --------------------------------------     =      ---------------------------------                                                             0.08/12                                           0.006667

                                              50{(1.006667)60 - 1}               50{1.489875 - 1}
                                      =    ------------------------------    =        -----------------------
                                                       0.006667                                0.006667

                                      =   50 x 0.489875                    =    24.49375    
                                               0.006667                               0.00667  
                                                                                  =   3673.87 

                                                                        Approx.  $3,673.87

Decay: The value of a certain machine has an annual decay factor of 0.75. If the machine is worth $10,000 after 5 years, what was the original value of the machine?

The machine will decay exponentially as in radioactive elements.
Therefore, applying the Exponential Decay Formula:
                                                    y = abx
Where y = the final amount, a = the original value, b the decay factor, 
and x the time in years.
                           By substitution, 10,000 = (a)(0.75)5
                                                       10,000 = (a)(0.2373)
                                                                 a =  10000
                                                                        0.2373
                                                                    =  42,140.75
                                                                        Answer: $42,140.75

Inequalities: Solve 3y + 4 > 2(y + 3) + y
             3y + 4 - 4 > 2y + 6 + y - 4  (Subtracting 4 from each side.)
                  3y + 0 > 2y + y + 6 - 4  (Grouping like terms.)
                        3y > 3y + 2          
                 3y - 3y > 3y -3y + 2  (Subtracting 3y from both sides.)
                         0 > 0 + 2

Answer: Since y is eliminated, there are no values to display

 


Trigonometry: Solve the equation, 2sinθ - sinθ = 0
                      This is a Quadratic Equation as in 2x - x = 0
                                    By factorization, Sinθ(2sinθ - 1) = 0
                                             So, sinθ = 0 and 2sinθ - 1 = 0 
            When sinθ = 0, angle θ = 0, 180 and 360
  Also, for 2sinθ - 1 = 0,  sinθ = 0.5, Angle θ = 30 and 150 

 Answer: Solutions of θ are 0, 30, 150, 180 and 360.

 

Radicals: Simplify √50 + √2 - 2√18 + √8.
    
 50 + √2 - 2√18 + √8 = √(25 x 2) + √2 - 2√(9 x 2) + √(4 x 2)
                                    = 5√2 + √2 - (2 x 3)√2 + 2√2 
                                    = 6√2 - 6√2 + 2√2 
                                   = 2√2
                                  Answer: 
2√2

 

Series: The fourth term of a Geometric Progression (G.P.) is -6 and the seventh term is 48. What are the first three terms of the progression?
                    
4th Term:  ar3 = -6  - equation 1
                     7th Term: ar= 48  - equation 2
 Dividing equation 2 by equation 1:
                                         ar= 48
                                 
    ar3 = -6
                                           r3 = -8
                                            r  = -2
 Substituting the value of r in any equation, say eq.2 to find a:
                                     a(-2)
  =  -6
                                        -8a  =  -6
                                           a  =  -6/-8  = 
 
 First Term of the GP is 

 Second Term is 
 X -2 = -6/4 = -1 and Third Term is -1 X -2 = 6/2 = 3
 
First three terms are 
,  -1,  3


Area Under A Curve: Find the area enclosed between the two curves y = 4 - x and y = x - 2x. 
     Where the two curves meet, x - 2x = 4 - x
                            Therefore, 2x - 2x - 4  =  0
                            Factorizing 2x - 2x - 4 = 0 to find the x coordinates,
                                              (x - 2)(x + 1) = 0
                                                                 x =  -1 and 2  
(the limits of integration)
   Required area between the two curves is
 {(4 - x)-(x - 2x)}dx, within the limits of -1 and 2.  (the Integral of the function)
                                                    
{(4 - x)-(x - 2x)}dx = (4 + 2x - 2x)dx  
                                                                                      = [4x + x - ⅔x] within -1 and 2
                                        When x = 2, [4x + x - ⅔x] = 20/3
                                       When x = -1, [4x + x - ⅔x] = -7/3 
Area under the curve = (20/3) - (-7/3) = (20/3) + (7/3) = 27/3 = 9

                                                             Answer: 9 Sq. Units

A graphical solution

 

Functions: Given the function f(x) = 2x - 5x + 3. 
Find the values of x and draw a graph of the function.

                   Let y = f(x) = 2x - 5x + 3
                      If y = f(x)
               Then y = 2x - 5x + 3

                         y = (x - 1)(2x - 3) by factorization
                When y = 0, x - 1 = 0 and 2x - 3 = 0
                    When x - 1 = 0, x = 1
                       When 2x - 3 = 0, x = 3/2 = 1
 

Values of x are 1 and 1

The solutions are the intercepts of the graph on
the x axis, 1 and 1.

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