State Competition - The Solutions
April 14, 2008 If 6 workers complete one-half of the project in 12 days then the 6 workers complete 1/(2 × 12) = 1/24 of the project in 1 day and one worker completes 1/(24 × 6) = 1/144 of the project in a single day.
Half of the project is done.
That means there is 1/2 = 72/144 of the project to go and this must be completed in 4 days.
Therefore, 18/144 of the project must be completed each day and we need 18 workers.
Four of those edges are just 10 cm each. To find the other sides, draw a “half” diagonal starting at where the center of the base to an edge (as I did in grey). That line is just half the diagonal of the 10 × 10 square. If d is the diagonal,
d2 = 102 + 102 = 100 + 100 = 200
d = √200 = 10√2 d/2 = (10/2)√2 = 5√2
Let x = the edge of the pyramid that is the hypotenuse of the triangle consisting of the line connecting the peak to the base and the half diagonal. Then
122 + (5√2)2 = x2
144 + 50 = 194 = x2
x ≈ 13.928388
13.928388 × 4 ≈ 55.713553 ≈ 56
56 + (4 × 10) = 56 + 40 = 96
Let x be the length of the jet.
Let y be the speed of the jet in feet per second. Remember that the jet must completely pass the hangar! Then x/y = 20
and (710 + x)/y = 70
x = 20y
710 + x = 70y
710 + 20y = 70y
710 = 50y
y = 710/50 = 71/5
x = 20y = (20 × 71)/5 = 71 × 4 = 284
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