Bosco

        A_sector  =  (83 / 360) • A_circle    

        A_circle  =  π r²    

        60 π  =  (83 / 360) • π r²      For the next step, cancel out the π's      

                                                     and divide both sides by (83 / 360)   

        r²  =  60 • (360 / 83)  =  260.24        

        r  =  sqrt(260.24)  =  16.13   

_.

Five times a number is divided by $2$ more than that number. If the result is $6,$ then what was the original number?  

5x / (x + 2)  =  6  

5x               =  6 • (x + 2)   

              5x  =  6x + 12   

             – x  =  12   

                 x  =  – 12    

_.

How many ways can you write two different letters chosen from  
A B C D E F G H I J K L M N O P  
so that they are in alphabetical order, and the first letter is a vowel?  

    34 ways  

If A is the first letter, then 15 possibilities    

If E is the first letter, then 11 possibilities    

If  I  is the first letter, then   7 possibilities    

If O is the first letter, then   1 possibility     

Add them up.  

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.  

The length of the median is equal to the average of the two bases.  

So, with one base 12, and the other base 16 longer, we have  

                  12 + (12 + 16)        40  

median  =  ––––––––––––  =  ––––  =  20  

                             2                   2  

The short base has to be 12, because 12 can't be 16 greater than another positive number.  

The area of the trapezoid has no bearing on the solution to the problem.  

.

I have $3$ different mathematics textbooks, $2$ different psychology textbooks, and $2$ different chemistry textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row?  

                 7!             5040  

₇P₇  =  –––––––  =  ––––––  =  5040       

             (7 – 7)!           1   

You might be alarmed by seeing zero in the denominator,   

which is the most verboten condition known in math,   

but zero factorial is defined as 1 so it's okay.  

_.

Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to $7$ times the largest of the seven integers. What is the smallest integer that Pearl wrote down?   

Call the smallest integer x.    

  (x) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6)  =  7 • (x+6)       

                                                                   7x + 21  =  7x + 42   

I don't see how you can get a solution to the problem as stated.   

_.

Find the vertex of the graph of the equation y = -2x^2 + 8x + 15 - 3x^2 + 7x - 25   

Combine like terms                                        y  =  –5x² + 15x – 10    

Take the first derivitive                                   y'  =  –10x + 15   

The first derivitive defines the slope of the   

curve at any point.  When the derivitive is   

equal to zero, that's where the parabola   

turns around and goes the other way.   

Set equal to zero, solve for x                          –10x + 15  =  0   

                                                                             x  =  –15 / –10  =  3 / 2  

This is the x-coordinate of the vertex    

Plug it in to the original equation                    y  =  –5x² + 15x – 10    

This is to see what y is when x=3/2    

                                                                       y  =  (–5)(3/2)² + (15)(3/2) – 10   

                                                                       y  =  (–5)(9/4) + (15)(3/2) – 10    

                                                                                –45 + 90 – 40        

Find the common denominator                       y  =  –––––––––––––  =  5 / 4         

                                                                                          4        

This is the y-coordinate of the vertex    

Put them together and the vertex is at (3/2 , 5/4)   or, if you prefer, (1.50 , 1.25)    

I think there's a formula for finding the vertex without using the first derivitive   

but I don't know what it is.  Maybe somebody else will chime in.   

_.