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?

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22-06-2021
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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?

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hoangkhangpham94 hoangkhangpham94 · Answer 1 · 2021-06-27T08:33:52+02:00

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%.