PROBLEMS INVOLVING DISTANCE AND THE PYTHAGOREAN THEOREM
The Pythagorean Theorem
The Pythagorean Theorem states the relationship between the lengths of the three sides of a right triangle:
C2 = A2 + B2, where A and B are the lengths of the two shorter sides (the legs) and C is the length of the longer side (the hypotenuse).
The CIRCUMFERENCE of a circle
The distance around a circle is called its circumference, and is determined by the circle's radius or diameter:
EXAMPLE 3.7.1
Find the missing side length for each triangle shown below.
see solutions
EXAMPLE 3.7.2
The diagram below shows the rectangular pen in which Gomer confines his wolverines and badgers. In order to prevent the wolverines from dating the badgers, Gomer is going to build a fence running from one corner of the pen to the opposite corner, thus dividing the pen into two smaller pens. Assuming that construction of such a fence will cost $1.25 per foot, find the total cost of this fence.
A. $4500
B. $500
C. $168
D. $56
see solution
EXAMPLE 3.7.3
Study the race course shown below. If Gomer runs 62 laps around the course, how many miles will he have run?
A. 4.6 miles
B. 9.2 miles
C. 7.5 miles
D. 4.4 miles
See solution
EXAMPLE 3.7.4
The diagram below shows the path the Plato takes when he goes for a philosophical stroll.
Plato starts at home, proceeds to the toga shop, then heads north to the tunic store, then returns home. On average, Plato thinks one profound thought for every 10 yards that he walks. Find the total number of profound thoughts that he will think during this walk.
A. 60
B. 10
C. 100
D. 258
E. None of these
see solution
EXAMPLE 3.7.5
The figure below shows an aerial view of The Hurl-O-Matic, a carnival ride in which the passengers are seated in a car, attached to the end of an arm which rotates rapidly around a central hub. Suppose that the length of the arm is 64 feet, and that, at full speed, it takes 10 seconds to for the car to complete one revolution. Find the speed of the car.
A. 40 miles per hour
B. 10 miles per hour
C. 27 miles per hour
D. 21 miles per hour
E. 37 miles per hour
See solution
EXAMPLE 3.7.6
Find the distance around the racetrack shown below.
A. 0.23 miles
B. 0.48 miles
C. 0.0067 miles
D. 0.35 miles
see solution
EXAMPLE 3.7.7
How fast (in miles per hour) does the Earth travel as it orbits the sun? (Note: the Earth's orbit is approximately circular, with a radius of 93,000,000 miles.)
EXAMPLE 3.7.8
The diagram on the left below shows the race course for the 40-K Wolverine Day Fun Run. The diagram on the right shows the course modified by the short-cut that Gomer uses. What distance does Gomer cover if he runs the race using his short-cut?
A. 36 km
B. 26 km
C. 10 km
D. 16 km
EXAMPLE 3.7.9
Plato and Aristotle are loitering on the street corner, when suddenly Socrates (to whom they owe money) shows up. Plato takes off running eastward at a rate of 16 miles per hour, and Aristotle runs southward at a rate of 12 miles per hour. How far apart (direct distance) are Plato and Aristotle after 15 minutes?
see solution
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