Bosco

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.   

64 x 3,816 = 244,224     

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Jon teaches a fourth grade class at an elementary school where class sizes are always at least 10 students and at most 15.  One day Jon decides that he wants to arrange the students in their desks in a rectangular grid with no gaps.  Unfortunately for Jon he discovers that doing so could only result in one straight line of desks.  What is the largest number of students that Jon could have in his class?    

It will have to be a prime number.  Any nonprime number can be divided into at least two rows.     

The largest prime number that's 15 or less is 13.    

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A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people, not including Mark?   

That depends.  If the two members have different ranks or titles, then there are ₁₄P₂  =  182 ways.    

This way would mean that a pair of members like Jesse and Frank is different from Frank and Jesse.    

If the two members are untitled and otherwise indistinguishable, then there are ₁₄C₂  =  91 ways.    

This way would mean that a pair of members like Jesse and Frank is the same as Frank and Jesse.  

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One ordered pair (a, b) satisfies the two equations ab^4 = 48 and ab = 24. What is the value of b in this ordered pair?    

                                                 ab⁴  =  48          (eq 1)    

                                                 ab    =  24          (eq 2)    

solve eq 2 for "a"                           a  =  24 / b    

sub "a" in eq 1                (24 / b) • b⁴  =  48      

divide both sides by 24             b⁴ / b  =  2    

                                                      b³  =  2    

                                                        b  =  cube root of 2    

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Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2 - 2x^2 + 4x - 1.    

When the quadratric is in the form ax² + bx + c the discriminant is b² – 4ac     

Rearranging terms so it's easier to see    5x² – 2x² – 2x² – 2x + 6x + 4x + 8 + 2 – 1     

                                                                   x² + 8x + 9    

                                                                    b² – 4ac  =  8² – (4)(1)(9)     

                                                                                    =  64 – 36    

                                                                                D  =  28    

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All members of our painting team paint at the same rate. If 20 members can paint a 2000 square foot wall in 40 minutes, then how long would it take the 20 members to paint a 6000 square foot wall, in minutes?    

Same number of painters, but three times the size of the wall.    

would take three times as many minutes:  3 • 40 min = 120 min    

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The parabola y = ax^2 + bx + c is graphed below.    

Find a + b + c. (The grid lines are one unit apart.)    
The parabola passes through (-2,8), (0,0), and (2,8).    

y = 2x² goes through the three specified points.   

Therefore, a+b+c = 2    

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Find the vertex of the graph of the equation y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25.    

first combine like terms      y  =  –5x² – 6x + 10    

take 1st derivitive               y'  =  –10x – 6    

and set equal to zero                  –10x – 6  =  0    

                                                   –10x        =  6    

                                                               x  =  –6/10 or –0.6     this is the x-coordinate    

                                                                                                 of the vertex    

substitute value of x    

into original equation           y  =  –5(–3/5)² – 6(–3/5) + 10    

                                            y  =  –5(9/25)   + 18/5     + 10    

                                            y  =  –9/5          + 18/5     + 10    

                                            y  =  59/5 or 11.8     this is the y-coordinate of the vertex     

The equation produces a downward opening parabola with its vertex at (–0.6 , 11.8)        

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My answer is that it cannot be done.   

First, I intuited this because both 52 and 39 are multiples of 13.   

Any subtaction of (52 • n1) – (39 • n2) will leave a multiple of 13.   

Second, I graphed 52x – 39y = 1 on Desmos.  I expanded the gridlines so that the  

distance between each large division was 1.  This was just to make it easier to read.  

I checked the path of the curve to see if it passes through any intersection of x and y  

perfectly, up to x = 20, because any x > 19 exceeds the 1,000 candies in the supply.  

The curve does not pass perfectly through any intersection.  This means that there  

is no solution that consists of two integers. 

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