Bosco

Sally used a pencil to trace the circular base of a drinking glass.  She placed four quarters in a line across the diameter of the circle.  Using a ruler, she measured the diameter of each quarter, and calculated the circumference of the glass as 29.78cm.   The quarter could be used as a referent  for a measure of one _______, and, according to Sally’s measurement, each quarter has a diameter of ________cm.      

C  =  π d     

29.78  =  3.14 • d     

d  =  29.78 / 3.14     

d  =  9.48     

4Q  =  d     

4Q  =  9.48     

  Q  =  9.48 / 4     

  Q  =  2.37     <<—— the diameter of one quarter     

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   What is the value of c if $x\cdot(3x+1)     

   In this context, c doesn't stand for an abbreviation of a number  .  

   \cdot is the LaTeX command used to produce a centered dot (⋅)     

   to indicate the multiplication function.     

   The example in the question could be written as x • (3x+1) if you     

   use Alt-0149 on the keypad to insert the dot instead of \cdot.     

   Read it as "x times the quantity three x plus one".     

_.     

The quadratic x^2 - 6x + 60 can be written in the form (x + b)^2 + c, where b and c are constants. What is b + c?     

In the instance of x² – 6x you need (6 / 2)² to complete the square.  So you have x² – 6x + 9     

                                                                                                                                 (x – 3)²     

To take that "9" on up to 60, simply add 51.  The resulting equation becomes (x – 3)² + 51     

Per the terms of the problem, add b + c, that is, add 3 + 51, and the final answer is 54.     

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 Five times a number is divided by 7 more than the original number.     

 If the result is 8, then what was the original number?     

     5X / (X+7)  =  8     

     5X  =  (8)(X+7)     

     5X  =  8X + 56     

  – 3X  =  56     

       X  =  – 56 / 3     

_.     

Let's write the problem like this:  In triangle ABC, with C=90^o, sin A = 4/5.  Find cos B.     

By Pythagoras, that is a 3-4-5 right triangle, thus the third side is 3.     

cosine equals adjacent over hypotenuse   ——>>   cos(B) = 4 / 5     

_.     

  Find cos C.     

  cosine equals adjacent over hypotenuse     

  cos(C)  =  7 / 25     

  cos(C)  =  0.2800     

_.     

What is the coefficient of x^2 when -5x^3 - 5x^2 - 7x + 1 is multiplied by -x^2 + 6x + 1 and the like terms are combined?     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>    – 5x²     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>  – 42x²     

   (–5x³ – 5x² – 7x + 1)(–x² + 6x + 1)  —–>>       – x²     

                combined total of coefficients          – 48     

   It isn't necessary to multply the whole thing out.     

   Just find only the terms that will contain an x².     

_.     

If (2x + 5)(x + 3) = 14, find the sum of the possible values of x.     

       (2x + 5)(x + 3)  =  14     

        2x² + 11x + 15  =  14     

        2x² + 11x + 1  =  0     

        It doesn't seem humanly possible to factor that quadratic, and     

        the quadratic formula involves too much arithmetic, so I used     

        Desmos to plot it, where I found that the two roots are     

                                              x  =  – 5.40754     

                                              x  =  – 0.09246     

                                       total        – 5.50000     

                        so, the sum is      – 5.5     

_.     

In triangle LMN, we have angle M = 90, MN = 2, and LM = sqrt(19). What is sin L?     

Use Pythagoras' Theorem to find that the hypotenuse is sqrt(23).     

sine = opposite over hypotenuse, therefore  sin(L) =  2 / sqrt(23)    ( 0.41703 )     

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