Mathematics, 31.12.2019 03:31, aredwolf2017

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?

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answered: hayden5928

Using an exponential function, we have that:

The growth rate is k = 0.101515

The function is:

$F(t) = 2.31e^{0.101515t}$

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

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:

$F(t) = F(0)e^{kt}$

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

Thus F(0) = 2.31 .

$F(t) = 2.31e^{kt}$

Item a:

In 2013, total federal receipts were $2.83 trillion, which means that F(2) = 2.83 , which is used to find k.

$F(t) = 2.31e^{kt}$

$2.83 = 2.31e^{2k}$

$e^{2k} = \frac{2.83}{2.31}$

$\ln{e^{2k}} = \ln{\frac{2.83}{2.31}}$

$2k = \ln{\frac{2.83}{2.31}}$

$k = \frac{\ln{\frac{2.83}{2.31}}}{2}$

k = 0.101515

The growth rate is k = 0.101515

The function is:

$F(t) = 2.31e^{0.101515t}$

Item b:

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

$F(4) = 2.31e^{0.101515(4)} = 3.47$

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:

$F(t) = 2.31e^{0.101515t}$

$10 = 2.31e^{0.101515t}$

$e^{0.101515t} = \frac{10}{2.31}$

$\ln{e^{0.101515t}} = \ln{\frac{10}{2.31}}$

$0.101515t = \ln{\frac{10}{2.31}}$

$t = \frac{ \ln{\frac{10}{2.31}}}{0.101515}$

t = 14.4

2011 + 14 = 2025.

Federal receipts will be of $10 trillion in 2025.

A similar problem is given at link

answered: proudmarinemom7186

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