Question

During 2012, global oil consumption grew by 0.7%, to reach 88 million barrels per day. Assume that it continues to increase at this rate, (a) Write the first four terms of the sequence a_n giving daily oil consumption n years after 2011; give a formula for the general term a_n. Round your answers for a_1, a_2, a_3, and a_4 to three decimal places.
a_1 =_____________.
a_2 = _____________.
a-3 = _____________.
a_4 = _____________.
a_n = _____________.
(b) In what year is consumption first expected to exceed 195 million barrels a day? Consumption exceeds 195 million barrels per day during the year

Answer 1

Answer:

(a)

$a_{1} = 88.000 \ million\\a_{2} = 94.160 \ million\\a_{3} = 100.751 \ million\\a_{4} = 107.804 \ million\\a_{n} = 82.243*1.07^n$

(b) 2024

Step-by-step explanation:

Global oil consumption in 2011 is given by:

$C_{2011} =\frac{88}{1.07}=82.243 \ million$

(a) Assuming a constant growth of 0.7% per year, the formula for the daily oil consumption n years after 2011 is:

$a_{n} = a_{0}*1.07^n\\a_{0} = C_{2011} = 82.243\\a_{n} = 82.243*1.07^n$

The terms a_1, a_2,a_3 and a_4, corresponding to the global oil consumption in the years of 2012, 2013, 2014 and 2015, respectively, are given by:

$a_{1} = 82.243*1.07^1\\a_{1} = 88.000 \ million\\a_{2} = 82.243*1.07^2\\a_{2} = 94.160 \ million\\a_{3} = 82.243*1.07^3\\a_{3} = 100.751 \ million\\a_{4} = 82.243*1.07^4\\a_{4} = 107.804 \ million$

(b) To find the in year in which consumption reaches 195 million barrels a day, apply logarithmic properties:

$a_{n} = 82.243*1.07^n\\ln(a_{n}) = ln(82.243)+ n*ln(1.07)\\\\n=\frac{ln(195)-ln(82.243)}{ln(1.07)} \\n= 12.760$

Consumption will reach 195 million barrels, 12.7 years after 2011, round it to the next whole year to find when consumption exceeds 195 million:

$Y = 2011+13 = 2024$