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The table shows the value of an account x years after the account was opened. Based on the exponential regression model, which is the best estimate of the value of the account 12 years after it was opened?

$8,910

$8,980

$13,660

$16,040

The table shows the value of an account x years after the account was opened Based on the exponential regression model which is the best estimate of the value o class=

Respuesta :

frika

Answer:

Correct choice is B

Step-by-step explanation:

The equation of the exponential regression model is [tex]y=a\cdot b^x.[/tex]

1. When x=0, [tex]y=a\cdot b^0=a=5,000;[/tex]

2. When x=2, then

[tex]y=5,000\cdot b^2=5,510\Rightarrow b^2=\dfrac{5,510}{5,000}=\dfrac{551}{500}=1.102,\ b=\sqrt{1.102}\approx 1.05.[/tex]

3. The equation of the function is [tex]y=5,000\cdot (1.05)^x.[/tex] Note that

  • [tex]y(5)=5,000\cdot (1.05)^5\approx 6,381\approx 6,390;[/tex]
  • [tex]y(8)=5,000\cdot (1.05)^8\approx 7,387\approx 7,390;[/tex]
  • [tex]y(10)=5,000\cdot (1.05)^{10}\approx 8,144\approx 8,150.[/tex]

Therefore,

[tex]y(12)=5,000\cdot (1.05)^{12}\approx 8,979\approx 8,980.[/tex]

Answer:

Option B is correct

$8,980

Step-by-step explanation:

The equation of exponent  regression is given by:

[tex]y=ab^x[/tex] , x is the time

where, a is the initial amount and b is the growth factor

As per the statement:

Let y represents the account value.

Given table as shown the value of an account x years after the account was opened.

Enter the values for x into one list and the values for y into the second list.    

Now, graph the scatter plot as shown below in the attachment.

then, we get the equation of exponent regression:

[tex]y = 4999.8 \cdot e^{0.0489x}[/tex]

We can write this as:

[tex]y = 4999.8 \cdot (e^{0.0489})^x[/tex]

⇒[tex]y = 4999.8 \cdot (1.05)^x[/tex]             ...[1]

We have to find the  best estimate of the value of the account 12 years after it was opened

Substitute x = 12 years in [1] we have;

[tex]y = 499.8 \cdot (1.05)^{12}[/tex]

⇒[tex]y = 4999.8 \cdot 1.7959[/tex]

Simplify:

y  ≈ $8979

Therefore,  the best estimate of the value of the account 12 years after it was opened is, $8980

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