Advanced Algebra II Notes 9.1 Using the Distance Formula

Investigation: Bucket Race The starting line of a bucket race is 5 m from one

end of a pool, the pool is 20 m long, and the finish line is 7 m from the opposite end of the pool, as shown. In this investigation you will find the shortest path from point A to point C on the edge of the pool to point B. That is, you will find the value of x, the distance in meters from the end of the pool to point C, such that AC + CB is the shortest path possible.

a) Make a scale drawing of the situation on graph paper. b) Plot several different locations for point C. For each, measure the distance x and find the total length AC + CB. Record your data.

X (m) AC (m) CB (m) AC + CB (m)

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c) What is the best location for C such that the length AC + BC is minimized? What is the distance traveled? Is there more than one best location? Describe at least two different methods for finding the best location for C. d) Make a scale drawing of your solution.

e) Now consider that you can carry an empty pail faster than you can carry a full bucket. You can carry an empty bucket at a rate of 1.2 m/s and a full bucket at a rate of .4 m/s.

x (m) Time for AC (s)

Time for CB (s)

total Time(s)

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212

212 )()( yyxxd

Distance Formula

Find the equation of the locus of points that are equidistance from the points (1, 3) and (5, 6).

An injured worker must be rushed from an oil rig 15 mi offshore to a hospital in the nearest town 98 mi down the coast from the oil rig. a) Let x represent the distance in miles from the point on the shore closest to the oil rig and another point, C, on the shore. How far does the injured worker travel, in terms of x, if a boat takes him to C and then an ambulance takes him to the hospital? b) Assume the boat travels at an average rate of 23 mi/h and the ambulance travels at an average rate of 70 mi/h. What value of x make the trip 3 hours?

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