Question

Have you ever heard of the Consumer Price Index (opens in a new window)? It measures the price level of certain goods and services in order to measure inflation. Suppose the Consumer Price Index is currently at 220 and is increasing at a rate of 3% per year. What will it be in 10 years? What exponential function could you set up to answer this question? This is an exponential growth problem with initial value If the rate is increasing at 3% each year, then the Index is staying the same but adding 3% each year 100% +3% 103% = 1.03 Therefore the index is growing at a rate of 103% each year and b = 1.03 This part is really important, remember that if your b value isn't at least 1, then you won't have an exponential growth function that is increasing like you're intending to A common mistake is to put 03 in as b instead of 1.03. Make sure when you are putting in your growth rate you are including the whole starting amount by adding 100% to your green rate before converting it to a decimal and using it as your b Using the general form of an exponential equation, y a(b) the function that represents this situation is y To find the value that the index will have in 10 years, we need to plug in 10 for z V- 220(1.03) 226(1.3430) 303,73 The value of the Consumer Price Index after 10 years will be about $ Search 13 2 a ?​

Answer 1

Explanation:

It seems like there might be some formatting issues in your message, but I understand the question. To model the Consumer Price Index (CPI) growth over time, we can use the exponential growth formula:

\[y = a \cdot b^x\]

where:

- \(y\) is the future value of the CPI,

- \(a\) is the initial value of the CPI,

- \(b\) is the growth factor (in this case, \(1 + \text{growth rate}\)),

- \(x\) is the time in years.

Given the information:

- Initial CPI (\(a\)) = 220,

- Growth rate = 3% or 0.03,

- Time (\(x\)) = 10 years,

we can set up the exponential function:

\[y = 220 \cdot (1 + 0.03)^{10}\]

Now, calculate the value:

\[y \approx 220 \cdot (1.03)^{10} \approx 303.73\]

So, the Consumer Price Index after 10 years is estimated to be about 303.73.