Respuesta :
Answer:
Step-by-step explanation:
An exponential equation can be written in the form
[tex]y=a*b^x[/tex]
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):
[tex]2.31=a*b^0[/tex] 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:
[tex]2.83=2.31b^2[/tex]
Divide both sides by 2.31 to begin to isolate the b:
[tex]\frac{2.83}{2.31}=b^2[/tex] and
[tex]b^2=1.225108225[/tex] 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:
[tex]y=2.31(1.106846)^x[/tex]
That answers part a.
For part b, we will use that equation to find y when x = 4 (2015-2011 = 4):
[tex]y=2.31(1.106846)^4[/tex]
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:
[tex]10=2.31(1.106846)^x[/tex]
Divide both sides by 2.31 to get:
[tex]4.329004329=1.106846^x[/tex]
Take the natural log of both sides to enable us to bring the x down front:
[tex]ln(4.329004329)=xln(1.106846)[/tex]
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.
Using an exponential function, we have that:
a)
The growth rate is [tex]k = 0.101515[/tex]
The function is:
[tex]F(t) = 2.31e^{0.101515t}[/tex]
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:
[tex]F(t) = F(0)e^{kt}[/tex]
In 2011 a country's federal receipts (money taken in) totaled $2.31 trillion.
Thus [tex]F(0) = 2.31[/tex].
[tex]F(t) = 2.31e^{kt}[/tex]
Item a:
In 2013, total federal receipts were $2.83 trillion, which means that [tex]F(2) = 2.83[/tex], which is used to find k.
[tex]F(t) = 2.31e^{kt}[/tex]
[tex]2.83 = 2.31e^{2k}[/tex]
[tex]e^{2k} = \frac{2.83}{2.31}[/tex]
[tex]\ln{e^{2k}} = \ln{\frac{2.83}{2.31}}[/tex]
[tex]2k = \ln{\frac{2.83}{2.31}}[/tex]
[tex]k = \frac{\ln{\frac{2.83}{2.31}}}{2}[/tex]
[tex]k = 0.101515[/tex]
The growth rate is [tex]k = 0.101515[/tex]
The function is:
[tex]F(t) = 2.31e^{0.101515t}[/tex]
Item b:
2015 is 4 years after 2011, thus this is F(4).
[tex]F(4) = 2.31e^{0.101515(4)} = 3.47[/tex]
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:
[tex]F(t) = 2.31e^{0.101515t}[/tex]
[tex]10 = 2.31e^{0.101515t}[/tex]
[tex]e^{0.101515t} = \frac{10}{2.31}[/tex]
[tex]\ln{e^{0.101515t}} = \ln{\frac{10}{2.31}}[/tex]
[tex]0.101515t = \ln{\frac{10}{2.31}}[/tex]
[tex]t = \frac{ \ln{\frac{10}{2.31}}}{0.101515}[/tex]
[tex]t = 14.4[/tex]
2011 + 14 = 2025.
Federal receipts will be of $10 trillion in 2025.
A similar problem is given at https://brainly.com/question/14747550
