There is a linear relationship between the number of cars (in vehicles per hour)

Question

There is a linear relationship between the number of cars (in vehicles per hour) on a freeway and the average speed (in miles per hour) of the cars. As the number of cars increases, the average speed decreases. When there are 1,000 vehicles per hour (vph) on the highway, the average speed is 78 mph. When there are 2,000 vehicles per hour (vph), the average speed is 62 mph. Answer the following questions.

a) What is the average speed when there are 1,714 vehicles per hour?

b) When the average speed is 69.1 mph, how many vehicles are on the highway?

Note: This is just a problem from high school algebra turned into a problem related to civil engineering. Set one variable as x and the other as y, find the slope, point-slope form, etc. and use that to answer the questions. Two points define a line, and you are given two points: (78 mph, 1000 vph) and (62 mph, 2000 vph). (25 points)

Explanation / Answer

a) let x denote vehicles per hour and y denote average speed

The two points are (1000,78) and (2000,62)

The equation of line passing through the two points is given by

(y-78)/(x-1000) = (62-78)/(2000-1000) = -0.016

y-78 = -0.016x+16

y = -0.016x+94

When x=1714, y = (-0.016*1714)+94 = 66.576 mph

Therefore, average speed will be equal to 66.576 mph

b)When y=69.1,

69.1 = -0.016x+94

x = 1556.25 vph

Therefore, there are 1556.25 vehicles per hour

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