Bosco

Does anybody else think this new AI interface is awful?   

I think it is, too.  I have a few math-related answers that   

have been waiting moderation/publishing for many days.   

But then, sometimes my answer goes up immediately.   

I think that the A.I. is a little short on the "I" part.  

Of course, you'll never see this comment if it languishes   

awaiting A.I. moderation for endless days.  :-)  

_.

How many integers from 1 to 9 are divisors of the five-digit number 24,551?   

This again?  The literal answer is that all of them are.   

But, of course, I think you mean which are evenly divisible into 24,551. 

This is easy to find out.  You just try all the digits that might qualify.   

It ends with an odd number, so you don't have to try any of the even numbers.   

It doesn't end with a 5 or a zero, so you don't have to try 5. 

So that leaves 3, 7, and 9 to try.  That shouldn't be too hard to do.  

edited to add:  Haha, I fell for it.  24,551 is a prime number; the answer is none.    

_.

Daniel filled 7/12 of a tank with 8 pails of water. Another 4 pails and 3 jugs are needed to fill the tank completely. How many jugs of water can the tank hold?  

8 pails of water brings the tank level up to 7/12.   

4 pails of water is half of 8 pails, so that raises the tank   

level up an additional half of 7/12 which is 7/24.   

Those 12 pails have taken the tank level to 7/12 + 7/24 = 21/24..   

The last 3 jugs of water adds the last 3/24 to completely fill the tank,   

therefore each jug is 1/24 of a tank.   

At 1/24 tank per jug, it would take 24 jugs to fill the tank.   

_.

Two men stand back-to-back and walk in opposite directions for $40$ yards each.  Each of them then turns left and walks another $40$ yards each.  In yards, how far are the two men from one another?  

square root of 80² + 80²  =  sqrt(12,800)  =  113.14 yards  

Draw it.  Remember, the men's lefts are opposite directions.      

The men are 80 yds north & south  

 and another 80 yds east & west. 

Just imagine that it's a big box, and solve for the diagonal.  

edited to add:  I calculated the diagonal by using the Pythagorean Theorem.  

Since it's a big square, we could have multiplied a side by the square root of 2.  

_.

We will use Pythagoras' Theorem to solve for AB.  

Actually, we won't even find AB, all we need is AB².  

Then we will use it again to solve for BC.  

AB²  =  17² – 7²  =  289 – 49  =  240    

BC²  =  AB² + AC²  =  240 + 3²  =  249   

BC  =  sqrt(249)  =  15.78    

_.

What is the area of a circle with circumference 16pi?  

Since the circumference is π • d, we can divide by π to get the diameter.  

The diameter, then, equals 16, which makes the radius half of that, i.e., 8.  

Area = π • r² so, substituting values, Area = 3.1416 • 8²  =  201.06 units²    

_.

Starting with the "2" add 1, then add 2, then add 4, then add 8, then add 16 .....   

so the next number will be 33.  To continue, keep doubling the amount that you add.   

_.

When the same constant is added to the numbers $60,$ $100,$ and $110,$ a three-term geometric sequence arises. What is the common ratio of the resulting sequence?   

In a geometric progression the ratio between succeeding numbers is the same.   

Therefore                                            100 + x         110 + x    

                                                            ––––––   =   ––––––     

                                                             60 + x          100 + x     

Cross multiplying                                 (x + 100)(x + 100)  =  (x + 60)(x + 110)    

                                                             x² + 200x +10,000  =  x² + 170x + 6600       

Subtract x² from both sides                  200x + 10,000  =  170x + 6600    

Subtract 170x from both sides                30x + 10,000  =  6600   

Subtract 10,000 from both sides                            30x  =  –3400       

                                                                     x  =  –113.333 • • •   

Plugging –113.333   

into the first ratio                                       –13.333   

                                                                 –––––––  =  0.24999  =  0.25   

                                                                  –53.333    

Plugging –113.333  

into the second ratio                                  –3.333   

                                                                 –––––––  =  0.24998  =  0.25    

                                                                  –13.333               

Who'd a thunk it would really work.  

The common ratio, after     

adding the constant, is                               0.25    

_.