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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Stopping Distance The police department must determine the speed limit on a bridge such that the flow rate of cars is maximum per unit time. The greater the speed limit, the farther apart the cars must be in order to keep a safe stopping distance. Experimental data on the stopping distances d (in meters) for various speeds v (in kilometers per hour) are shown in the table.

v 20 40 60 80 100
d 5.1 13.7 27.2 44.2 66.4

(a) Convert the speeds v in the table to speeds s in meters per second. Use the regression capabilities of a graphing utility to find a model of the form d ( s ) = a s 2 + b s + c for the data.

(b) Consider two consecutive vehicles of average length 5.5 meters, traveling at a safe speed on the bridge. Let T be the difference between the times (in seconds) when the front bumpers of the vehicles pass a given point on the bridge. Verify that this difference in times is given by

T = d ( s ) s + 5.5 s

(c) Use a graphing utility to graph the function T and estimate the speed s that minimizes the time between vehicles.

(d) Use calculus to determine the speed that minimizes T. What is the minimum value of T? Convert the requireds peed to kilometers per hour.

(e) Find the optimal distance between vehicles for the speed found in part (d).

(a)

To determine

To calculate: The quadratic regression model for the provided data after converting the speed into metres per second; v20406080100d5.113.727.244.266.4

Explanation

Given:

The data set for the distance d in metres and the speed v in kilometres per hour.

v20406080100d5.113.727.244.266.4

Calculation:

Consider v=20 km/hr. This can be converted to meters per second as follows:

20 km/hr=2036001000=5.56 m/s

Similarly,

40 km/hr=4036001000=11.11 m/s60 km/hr=6036001000=16

(b)

To determine

To prove: The difference in times when the vehicles passed a certain point is T=d(s)s+5.5s.

(c)

To determine

To calculate: The speed that minimizes the time between the vehicles where d(s)=0.071s2+0.389s+0.727

(d)

To determine

To calculate: The speed in kilometres per hour that minimizes the time between the vehicles and the minimum value of the time where d(s)=0.071s2+0.389s+0.727

(e)

To determine

To calculate: The optimal distance between two vehicles if d(s)=0.071s2+0.389s+0.727.

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