Bosco

Chris, you were right all along.  The problem stipulates x not equal to y. 

I had overlooked that in all that run-together text and those dollar signs. 

After I realized it, too late to edit, I was relieved that my comment got lost  

in the moderation by AI, which btw I maintain is a little short on the "I" part.   

30 + 30 is 60   

_.

A stick is broken at two points, chosen at random. If the length of the stick is 6 units, then what is the probability that all three resulting pieces are shorter than 2 units?  

Probability is zero.  If all 3 pieces are shorter than 2 units, then they don't add up to the original 6 of the stick.  

_.

prove that there is only one pair of (integer) perfect squares that differ by 53, and find it    

Let's call the squares x² and y² so                  y² – x²  =  53   

Well, I couldn't figure out how to do it, so   

I went to Desmos and plotted the curve of       y  =  sqrt(x² + 53)   

and by careful scrutinization, discovered  

the place where the curve crossed two  

whole numbers on the grid.  It wasn't    

as hard as describing it sounds.  

The answer is when x = 26 & y = 27               27² – 26²  =  53   

                                                                       729 – 676  =  53   

I can't prove that's the only pair.  Sorry.   

_.

Two points on a circle of radius $1$ are chosen at random. Find the probability that the distance between the two points is at most $1.5.  

The phrase "the distance between the two points is at most $1.5"   

can also be stated as "is within 1.5."  They both mean the same thing.  

The first point can be anywhere.  Then, we're looking for the probability  

that the second point is within 1.5 of the first point.  I'm assuming that the  

distance is measured around the circumference, and not straight across.  

The 1.5 can be either clockwise or counterclockwise, so there is a 3.0  

band that the second point can be in.  

The circle diameter is 2.0 so the entire circumference is 2 • π = 6.28.   

The probability that the two points are within the 3.0 band is 3.0 / 6.28.   

3.0 / 6.28 = 0.48   ...  times 100 to make it a percentage is 48% probability.  

_.

What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?      

Combine like terms and arrange in standard  

quadratic format like ax² + bx + c                         t² + 4t – 96    

The t² tells us the curve is a parabola, and the fact   

 that t² is positive tells us the parabola opens upward   

The first derivitive of the equation describes the slope of the curve  

The vertex is the place where the curve turns around and goes back up,  

therefore is the smallest value, and the slope is zero at the vertex       

The first deritive is                                               2t + 4  

Set it equal to zero and solve for t                       2t + 4 = 0   

                                                                                    t = –2   

This is the answer to "What real value of t" 

but let's keep going and find out just what   

that smallest value actually is   

Substitute –2 for t back into original equation        (–2)² + 4(–2) – 96    

                                                                                   4 – 8 – 96 = –100   

So the smallest value is –100 which occurs when t = –2    

_.

How many positive integers are not less than the sum of their divisors?      

I'd make the argument that any integer can be divided by 1 and by itself.  

From this, we can say that the sum of these divisors of any integer x is x+1. 

Therefore, every integer is less than this particular sum of its divisors. 

Except the integer 1.  1 is divisible only by 1, so the integer is the same as the sum. 

So, I conclude that the answer is one ... namely, the integer 1.  

_.

1 + \frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{2}{25} + \frac{3}{25} + \frac{4}{25}   

           1         2          3           2         3           4  

  1 + –––– + –––– + –––– + –––– + –––– + ––––   

           5         5          5          25       25         25            

  You have to get everything with a common denominator.  

    25        5         10         15         2          3         4  

  –––– + –––– + –––– + –––– + –––– + –––– + ––––  

    25       25         25        25        25        25        25            

    64  

  ––––   =   2 & 14 / 25    

    25   

_.

                     pole height   

tan(72)  =  ––––––––––––   

                          45    

                        pole height   

3.077684  =  ––––––––––––   

                            45    

pole height  =  45 • 3.077684   

pole height  =  138.496 to the nearest foot is 138    

_.