The following table shows a company's annual income over a 6-year period. The equation y=60000(1.2)x describes the curve of best fit for the company's annual income (y). Let x represent the number of years since 2001.
Answer :
Given that the annual income of a company over a 6-year period is described by the equation:
[tex]\begin{gathered} y=60000(1.2)^x \\ \text{where} \\ x\text{ is the number of years since 2001} \end{gathered}[/tex]The annual income at the end of each year since 2001 is as shown in the table below:
Required: To evaluate the company's approximate annual income in 2009.
Solution:
Given the annual income described as
[tex]y=60000(1.2)^x[/tex]The number of years between 2001 and 2009 is evaluated as
[tex]x\text{ = 2009 -2001 = 8 years}[/tex]thus, it's been 8 years since 2001.
The annual income in 2009 is thus evaluated by substituting 8 for the value of x in the annual income function.
This gives
[tex]\begin{gathered} y=60000(1.2)^x \\ x\text{ = 8} \\ \text{thus,} \\ y\text{ = 60000}\times(1.2)^8 \\ =\text{ 60000}\times4.29981696 \\ y=\text{ }257989.0176 \\ \Rightarrow y\approx258000 \end{gathered}[/tex]Hence, the company's approximate annual income in the year 2009 will be $ 258000.
The third option is the correct answer.
The company's approximate annual income for the year 2009, using the given exponential growth model, would be $257,989.02.
The closest option to this calculated value is $258,000.
The correct option is (C).
To find the company's approximate annual income for the year 2009 using the given equation:
1. Identify the given information: The equation for the annual income is [tex]\( y = 60000 \times (1.2)^x \)[/tex], where ( x ) is the number of years since 2001.
2. Calculate the value of ( x ): Since we want the income for the year 2009, ( x ) would be 2009 - 2001, which is 8.
3.Substitute the value of ( x ) into the equation to find ( y ), the approximate annual income for the year 2009.
Let's perform the calculation:
[tex]\[ y = 60000 \times (1.2)^8 \][/tex]
Now, we'll compute the value using the provided equation.
The company's approximate annual income for the year 2009, using the given exponential growth model, would be $257,989.02.
The closest option to this calculated value is $258,000.
