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Answer:
Step-by-step explanation:
An exponential equation can be written in the form
where a is the initial condition and b is the growth/decay rate. It's growth if the number is greater than 1 and it's decay if the number is less than 1 but greater than 0. Our initial conditions are, in coordinate form where 2011 is year 0:
(0, 2.31) (2, 2.83)
Use those to find the equation for our situation. Always start out with the coordinate that has a 0 as x, since anything raised to the power of 0 is equal to 1. So we begin with the coordinate (0, 2.31):
so
2.31 = a
Now we use that a value along with the x and y from the other coordinate to come up with our growth or decay rate:
Divide both sides by 2.31 to begin to isolate the b:
and
so
b = 1.106846
That means that this is a growth rate (but you knew that already).
The equation then for this particular situation is:
That answers part a.
For part b, we will use that equation to find y when x = 4 (2015-2011 = 4):
Raise what's inside the parenthesis to the power of 4 to get:
y = 2.31(1.500889778) so
y = 3.47 trillion
For part c. we are looking for x when y = 10:
Divide both sides by 2.31 to get:
Take the natural log of both sides to enable us to bring the x down front:
Now divide both sides by the growth rate to get that
x = 14.434 or approximately 14 years after 2011 which is the year 2025.