1. Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.
  2. The Hudson River flows at a rate of 3 miles per hour.  A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours.  What is the speed of the boat in still water?
  3. A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
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1 Answer

1. The time in hours, t=200/s, where s=speed. t-1=200/(s+10). So (t-1)(s+10)=200 and ts+10t-s-10=200. Therefore 10t-s=10 because st=200 causes 200 to cancel out. 10*200/s-s=10 and 2000-s^2=10s; s^2+10s-2000=0. This can be factorised: (s+50)(s-40)=0, so s=40 mph, since negative speeds can be ruled out.

2. Let s=speed of the boat. The actual speed of the boat when travelling against the current is s-3 and with the current is s+3. The distance is 60 miles. The total time is 60/(s-3)+60/(s+3)=9. Multiply through by s^2-9: 60(s+3+s-3)=9s^2-81. 120s=9s^2-81, 3s^2-40s-27=0. Using the quadratic formula: s=(40+sqrt(1600+4*3*27))/6=(40+sqrt(1600+324))/6=(20+sqrt(481))/3, giving s=13.98 mph, because the negative answer is rejected.

3. If the number of rows and columns is n, then n^2=c-24, where c=number of characters. (n+1)^2=c+25, because there is a surplus of 24 characters when they are arranged in n twos and columns, and a deficiency of 25 when they are arranged in n+1 rows and columns. (n+1)^2-n^2=2n+1=c+25-(c-24)=49, so 2n=48 and n=24. c-24=n^2=576, so c=600. The number of characters is therefore 600.

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