The United States has been consuming iron ore at the rate of R (t) million metric tons per year at time t, where t is measured in years since 1980 (that is, t = 0 corresponds to the year 1980), and R (t) = 18.94e0.02t Find a formula T (t) for the total U. S. consumption of iron ore, in millions of metric tons, from 1980 until time t T (t) =

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)