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 (tequals0).
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?

Using an exponential function, we have that:

a)

The growth rate is

The function is:

b)

The estimate for the total federal receipts in 2015 is of $3.47 trillion.

c)

Federal receipts will be of $10 trillion in 2025.

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Assume that the growth of federal receipts, F, can be modeled by an exponential function and use 2011 as the base year:

This means that the function for the amount in t years after 2011, in trillions of dollars, is:

In 2011 a country's federal receipts (money taken in) totaled $2.31 trillion.

Thus .

Item a:

In 2013, total federal receipts were $2.83 trillion, which means that , which is used to find k.

The growth rate is

The function is:

Item b:

2015 is 4 years after 2011, thus this is F(4).

The estimate for the total federal receipts in 2015 is of $3.47 trillion.

Item c:

This is t years after 2011, considering t for which F(t) = 10. Thus:

2011 + 14 = 2025.

Federal receipts will be of $10 trillion in 2025.

A similar problem is given at link

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.

ok

step-by-step explanation: