Question

In 2011 a​ country's federal receipts​ (money taken​ in) totaled ​$2.31 trillion. In​ 2013, total federal receipts were ​$2.83 trillion. Assume that the growth of federal​ receipts, F, can be modeled by an exponential function and use 2011 as the base year ​(tequals​0).
​a) Find the growth rate k to six decimal​ places, and write the exponential function​ F(t), for total receipts in trillions of dollars.
​b) Estimate total federal receipts in 2015. ​
c) When will total federal receipts be ​$10 ​trillion?

Answer 1

Answer:

Step-by-step explanation:

An exponential equation can be written in the form

$y=a*b^x$

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):

$2.31=a*b^0$ 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:

$2.83=2.31b^2$

Divide both sides by 2.31 to begin to isolate the b:

$\frac{2.83}{2.31}=b^2$ and

$b^2=1.225108225$ 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:

$y=2.31(1.106846)^x$

That answers part a.

For part b, we will use that equation to find y when x = 4 (2015-2011 = 4):

$y=2.31(1.106846)^4$

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:

$10=2.31(1.106846)^x$

Divide both sides by 2.31 to get:

$4.329004329=1.106846^x$

Take the natural log of both sides to enable us to bring the x down front:

$ln(4.329004329)=xln(1.106846)$

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.