i.
Within a major United States city, the annual tonnage of solid
waste (garbage) has been increasing at an exponential rate of 10
percent per year. Assume that the current daily tonnage is 50000
tons and the rate and pattern of growth continue.
What daily tonnage will be expected 15 years from now?
b. Current capacity for handling solid waste is 8000 tons per day.
When will this capacity be no longer be sufficient?
a.
Answers
Given : the annual tonnage of solid waste (garbage) has been increasing at an exponential rate of 10 percent per year.
current daily tonnage is 5000 tons
To Find : What daily tonnage will be expected 15 years from now
Current capacity for handling solid waste is 8000 tons per day.
When will this capacity be no longer be sufficient?
Solution:
10 % = 0.1
Let say units = x
Then dx /dt = 0.1x ( t is in years)
=> dx/ 0.1x = dt
integrating both side
(1/0.1) ln (x) = t + c₁
=> ln (x) = 0.1t + c
at t = 0 , x = 5000
ln (5000) = + c
=> c = 8.517
ln (x) = 0.1t + 8.517
t = 15
=> ln (x) = 0.1 * 15 + 8.517
=> ln (x) = 10.017
=> x = 22,404
ln (8000) = 0.1 * t + 8.517
0.47 = 0.1 * t
=> t = 4.7
Hence after 4.7 years capacity be no longer be sufficient
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