Respuesta :
Answer:
T(t) = 947*(e^0.02t - 1)
Step-by-step explanation:
Given:
R(t) = 18.94 e^0.02t
Find:
Find a formula T(t) for the total U.S. consumption of iron ore, in millions of metric tons, from 1980 until time t.
Solution:
- The rate of change:
dT / dt = R(t)
dT / dt = 18.94 e^0.02t
- Separate variables:
dT = 18.94 e^0.02t .dt
- Integrate both sides:
T = (18.94/0.02)*e^0.02t + C
T = 947*e^0.02t + C
- Evaluate C where T = 0 @ t = 0:
0 = 947*1 + C
C = -947
- Hence,
T(t) = 947*(e^0.02t - 1)
Using a integral, it is found that the formula for total U.S. consumption of iron ore, in t years after 1980, is:
[tex]T(t) = 947e^{0.02t} - 947[/tex]
The yearly consumption, in t years since 1980, is modeled by:
[tex]R(t) = 18.94e^{0.02t}[/tex]
The total is the integral of the yearly consumption, hence:
[tex]T(t) = \int_{0}^t R(t) dt[/tex]
[tex]T(t) = \int_{0}^t 18.94e^{0.02t} dt[/tex]
[tex]T(t) = 947e^{0.02t}|_{0}^{t}[/tex]
[tex]T(t) = 947e^{0.02t} - 947[/tex]
A similar problem is given at https://brainly.com/question/21439037
