Math Problem Statement

If the rate of inflation is 2.8% per year, the future price p(t) (in dollars) of a certain item can be modeled by the following exponential function: p(t) = 2500 * (1.028)^t, where t is the number of years from today. Find the price of the item 5 years from today and 8 years from today. Round your answers to the nearest dollar as necessary.

Solution

To solve the problem, we use the given exponential function for the price:

p(t)=2500(1.028)tp(t) = 2500 \cdot (1.028)^t

Step 1: Calculate the price 5 years from today (t=5t = 5).

Substitute t=5t = 5 into the function:

p(5)=2500(1.028)5p(5) = 2500 \cdot (1.028)^5

Perform the calculation:

p(5)=25001.147782869p(5) = 2500 \cdot 1.14778 \approx 2869

Step 2: Calculate the price 8 years from today (t=8t = 8).

Substitute t=8t = 8 into the function:

p(8)=2500(1.028)8p(8) = 2500 \cdot (1.028)^8

Perform the calculation:

p(8)=25001.241863105p(8) = 2500 \cdot 1.24186 \approx 3105

Final Answer:

  • Price 5 years from today: $2869
  • Price 8 years from today: $3105

Would you like me to explain the steps further or assist with another calculation?


Here are 5 related questions for further exploration:

  1. What would the price be 10 years from today?
  2. How does the price change if the inflation rate is increased to 3.5%?
  3. What is the percentage increase in the item's price between 5 and 8 years?
  4. Can the formula be used to calculate the price for fractional years (e.g., 2.5 years)?
  5. What would the original price (p(0)p(0)) have been 10 years ago if the inflation formula were reversed?

Tip: When dealing with exponential growth problems, always check how the percentage growth affects compounding over time!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponential Growth
Inflation
Compound Interest

Formulas

p(t) = P * (1 + r)^t

Theorems

Exponential Growth Formula

Suitable Grade Level

Grades 9-12