Question: The annual amount of crude-oil production in a country (in millions of barrels) can be approximated…

The annual amount of crude-oil production in a country (in millions of barrels) can be approximated by the function \( f(t) = 1743(1.039)^t \), where \( t = 8 \) corresponds to the year 2008.

(a) Find the amount of production in 2014.

(b) If the trend continues, find the amount of production in 2026.

Solution

The function given is \( f(t) = 1743(1.039)^t \), where \( t = 8 \) corresponds to the year 2008. ### (a) Find the amount of production in 2014. First, find \( t \) for the year 2014. \[ t = 2014 - 2008 + 8 = 14 \] Plug \( t = 14 \) into the function. \[ f(14) = 1743(1.039)^{14} \] Calculate \( (1.039)^{14} \). \[ (1.039)^{14} \approx 1.60471 \] Now, multiply by 1743. \[ f(14) \approx 1743 \times 1.60471 = 2797.01 \] So, the estimated amount of production in 2014 is approximately 2797 million barrels. ### (b) Find the amount of production in 2026. First, find \( t \) for the year 2026. \[ t = 2026 - 2008 + 8 = 26 \] Plug \( t = 26 \) into the function. \[ f(26) = 1743(1.039)^{26} \] Calculate \( (1.039)^{26} \). \[ (1.039)^{26} \approx 2.37978 \] Now, multiply by 1743. \[ f(26) \approx 1743 \times 2.37978 = 4144.74 \] So, the estimated amount of production in 2026 is approximately 4145 million barrels.