CPhill

Heureka, I see what you did with that algorithm to find the relative primes to "n.'   I've not seen that done before........but....it's pretty neat.......is there a proof of this????

I assume you want to know how to access the tangent inverse function.....just hit the "2nd" key and you will see the "atan" function key ......  (tangent inverse key)

Let's call the time it takes the first car to travel the dstance, t.

Then, the time the second car travels is (t - 3.7) hrs.    ..... and since the distances are the same, we have

54(t - 3.7)  = 41t      simplify

54t - 199.8  =  41t     rearrange

54t - 41t = 199.8      simplify

13t = 199.8        divde both sides by 13

t =  about 15.369 hrs

Notice that the sum of the first n terms and the next term to be added are relatively prime, since their difference is just 1.  For instance:    1  + 2  + 4  = 7.  And the next term to be added is 8.

But, if A, B are relatively prime, then their sum, A + B, (the next  term in the series), must be relatively prime to B, because their difference is just A. [If A and B weren't relatively prime, there would be some factor, k, that would divide them both and would also divide their sum.]

Thus, each successive term in the series is relatively prime to its successor........

Here's another way to do this based on Alan's drawing...

The hypotenuse of the the small triangle at the bottom left = √[8^2 + 8^2] = 8√2

So, the length of the hypotenuse of larger triangle at the top right is just 27 - 8√2

And using similar triangles, the length of each side of this larger triangle = w =  [27 - 8√2]/ √2  = about 11.092 ft.  ≈  11.1 ft

33)    0.25/2ft = x/15ft    where x is the inches on the scale drawing   multiply by 15 on both sides

(15)(0.25)/ 2 = x  = 1.875 in

34)   The tiled area  = Total area - area occupied by cabinets.......

The total area is 15 x 12 =  180 sq ft

The area occupied by the cabinents is (12 x 2)  + (4 x 2 x 2)  =   24 + 16 = 40 sq ft

So  ...   180 - 40   =  140 sq ft  will be tiled (not covered by cabinets)

35) I'm taking a guess here as to what is meant......

10 cabinets will be installed.....so......the cost per cabinet is  $(2150 - 650)/ 10   =  $150  per cabinet

So  20 cabinets plus the fixed labor charge of $650  =  20 x $150 + $650  = $3650

(22/6) = 11/3  ....so....

11/3 gallons uses per hour x (1/3) hr  = 11/9 gallons used = 1 + 2/9   gallons used

Melody.....could you explin how that last conversion to base 16 works????........I see what you did, but I can't quite wrap my head around that.....