hw 10 question 2

Question

In 1990 (t = 0), the world use of natural gas was 74136 billion cubic feet, and the demand for natural gas was growing exponentially at the rate of 6.5% per year. If the demand continues to grow at this rate, how many cubic feet of natural gas will the world use from 1990 to 2017?

Answer 1

This problem seems to require a little thinking. When told that the usage is 74,136 billion feet in 1990, my question is 74,136 billion cubic feet per what? (like "per second", "per minute", per hour", per what?) The units don't tell us so we must assume that 74,136 billion cubic feet was used per year (the year of 1990). That means to find the total amount of natural gas used between 1990 and 2017, we must integrate the rate function.

So let's first create a rate function that calculates natural gas usage rate as a function of time - we'll call natural gas usage rate "N", so we are looking for N(t). Using our knowledge of exponential functions, that would be:

N(t) = 74136(1 + 0.065)^t or

N(t) = 74136(1.065)^t where N(t) is in units of "billion cubic feet per year" and "t" is in units of "years after 1990".

So, since 2017 is year 27, the total amount of gas used is ₀∫²⁷N(t) dt =

₀∫²⁷74136(1.065)^t dt = 74136•₀∫²⁷ 1.065^t dt

At this point, you can actually just plug this into a calculator to get the answer. To do it manually, it is helpful to think about the exponential derivative rule.

So the antiderivative is 74136•(1.065^t)/ln(1.065) evaluated between 0 and 27 = 74136/ln(1.065)[1.065²⁷ - 1.065⁰] = 5,268,937 billion cubic feet of natural gas or 5.269 x 10¹⁵ cubic feet