In the year $2002(t=0)$, the world reserves of natural gas were approximately 9671 trilion cubic feet. In that same year, the world consumption of natural gas was approximately 97.9 trillion cubic feet, and was growing exponentially at about $1.9 \%$ per year. If the demand continues to grow at this rate, and no new reserves of natural gas are found, in what year will the world reserves of this resource be depleted?
The world reserves will be depleted in the year (Round to the nearest year as needed.)
Answer
Adding this to the initial year of 2002, we find that the world reserves of natural gas will be depleted in the year \(\boxed{2058}\) (rounded to the nearest year).
Step 1 ：In the year 2002, the world reserves of natural gas were approximately 9671 trillion cubic feet. In that same year, the world consumption of natural gas was approximately 97.9 trillion cubic feet, and was growing exponentially at about 1.9% per year. If the demand continues to grow at this rate, and no new reserves of natural gas are found, we want to find the year when the world reserves of this resource will be depleted.
Step 2 ：We are given the initial amount of natural gas reserves, the initial consumption rate, and the growth rate of consumption. We can model the consumption as an exponential function, and the reserves as a linear function (since they are being depleted at a rate proportional to the consumption).
Step 3 ：We want to find the year when the reserves will be depleted, i.e., when the reserves reach zero. This is equivalent to finding the time when the total consumption equals the initial reserves.
Step 4 ：The total consumption at time t (in years) can be calculated as the integral of the consumption rate from 0 to t. The consumption rate at time t is given by the initial consumption rate times the exponential of the growth rate times t. Therefore, the total consumption at time t is the integral of this function from 0 to t.
Step 5 ：We can solve this problem by setting up an equation for the total consumption at time t, setting it equal to the initial reserves, and solving for t.
Step 6 ：By solving the equation, we find that the time when the total consumption equals the initial reserves is approximately 55.61658238 years from 2002.
Step 7 ：Adding this to the initial year of 2002, we find that the world reserves of natural gas will be depleted in the year \(\boxed{2058}\) (rounded to the nearest year).
