Word problem on Function and gas concentration as its dependencies

Functions are denoted as shown in the figure where there will be dependencies of one variable value to another.

Topic : Functions and variables values.

Variables, co-efficient are the terms found in functions.

Problem : The table lists future concentrations of a certain green house gas in parts per billion (ppb), If current trends continue.

a) Let x = 0 correspond to 2010 and x = 20 correspond to 2030. Find C and a so that f(x) = Ca^x models the data.

b) Estimate the gas concentration in 2021.

Year ------ Gas(ppb)
2010 ------ 0.74 (when x = 0)
2015 ------ 0.86
2020 ------ 0.99
2025 ------ 1.15
2030 ------ 1.34 (when x = 20)

Solution :

a) We have function ,
f(x) = Ca^x

x = 0 ; f(0) = Ca⁰ = C*1 = C

So f(0) = 0.74 ; C = 0.74

and x = 20 ; f(x) = C a²⁰ = 0.74 a²⁰

f(20) = 1.34 (from the table)

So 1.34 = 0.74 a²⁰ ; a²⁰ = 1.34/0.74 = 1.8108

So a = 1.0301


b) To find gas concentration at the year 2021, we have to find f(21)

f(21) = Ca²¹ = 0.74 (1.0301)²¹

= 0.74 * 1.8641

= 1.37943

So f(21) approximately 1.38

Therefore gas concentration at 2021 will be 1.38

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