The table below shows the average value of houses in Wilshire Woods over a period of 5 years. The equation [tex]f(x)=20,000 \cdot 3^x[/tex] describes the curve of best fit for the average value of houses [tex]f(x)[/tex]. Let [tex]x[/tex] represent the number of years since 2000, so [tex]x=0[/tex] in 2000.
| Year | Average Home Value |
| --- | --- |
| 2000 | $19,000 |
| 2001 | $60,000 |
| 2002 | $182,000 |
| 2003 | $540,000 |
| 2004 | $1,621,000 |
Using the equation for the curve of best fit, what will be the average home value in Wilshire Woods in 2005?
A. $36,450,000
B. $12,154,000
C. $4,860,000
D. $750,000
Answer (1)
Determine the value of x for the year 2005: x = 2005 − 2000 = 5 .
Substitute x = 5 into the equation f ( x ) = 20 , 000 × 3 x .
Calculate 3 5 = 243 .
Calculate f ( 5 ) = 20 , 000 × 243 = 4 , 860 , 000 . The average home value in 2005 is $4 , 860 , 000 .
Explanation
Understanding the Problem We are given the equation f ( x ) = 20 , 000 × 3 x that models the average home value in Wilshire Woods, where x is the number of years since 2000. We want to find the average home value in 2005.
Finding the Value of x To find the average home value in 2005, we need to determine the value of x for the year 2005. Since x is the number of years since 2000, we have x = 2005 − 2000 = 5 .
Substituting x into the Equation Now, we substitute x = 5 into the equation f ( x ) = 20 , 000 × 3 x to find the average home value in 2005: f ( 5 ) = 20 , 000 × 3 5
Calculating 3^5 We calculate 3 5 : 3 5 = 3 × 3 × 3 × 3 × 3 = 243
Calculating f(5) Now, we multiply 20,000 by 243: f ( 5 ) = 20 , 000 × 243 = 4 , 860 , 000
Final Answer Therefore, the average home value in Wilshire Woods in 2005 will be $4 , 860 , 000 .
Examples
Understanding exponential growth, as modeled by the equation in this problem, is useful in many real-world scenarios. For example, it can help in predicting population growth, the spread of a virus, or the increase in the value of investments over time. By using a mathematical model, we can make informed decisions about the future based on current trends.
