Question

Jeff, a local traffic​ engineer, has designed a new pedestrian foot bridge that is capable of handling the current traffic rate of 300 pedestrians daily. Once the traffic rate reaches 2 comma 000 pedestrians​ daily, however, the bridge will require a new bracing system. Jeff has estimated that traffic will increase annually at 3 ​%. How long will the current bridge system work before a new bracing system is​ required? What if the annual traffic rate increases at 8 ​% ​annually? At what traffic increase rate will the current system last only 12 ​years?

Answer 1

Answer:

a. How long will the current bridge system work before a new bracing system is​ required?: 64.18 years or 64 years and 2 months.

b. What if the annual traffic rate increases at 8 ​% ​annually: The bracing system will last for 24.65 years or 24 years and 7 months.

c. At what traffic increase rate will the current system last only 12 ​years: 17.13%

Explanation:

a. Denote x is the time taken for the number of pedestrian to grow from 300 to 2000. The current pedestrian is 300, the grow rate per year is 3% or 1.03 times a year. Thus, to reach 2,000, we have the equation: 300 x 1.03^x = 2000. Show the equate, we have 1.03^x = 6.67 <=> x = 64.18

b.  Denote x is the time taken for the number of pedestrian to grow from 300 to 2000. The current pedestrian is 300, the grow rate per year is 8% or 1.08 times a year. Thus, to reach 2,000, we have the equation: 300 x 1.08^x = 2000. Show the equate, we have 1.08^x = 6.67 <=> x = 24.65.

c. Denote x as traffic increase rate. The current pedestrian is 300, the grow rate per year is (1+x) times a year. Thus, to reach 2,000 after 12 years and thus a new bracing system to be in place, we have the equation: 300 x (1+x)^12 = 2000. Show the equate, we have (1+x)^12 = 6.67 <=> 1+x = 1.1713 <=> x = 17.13%.