Math Problem Statement

The function f(t) = 2100(0.25)^(t/12) represents the change in a quantity over t months. What does the constant 0.25 reveal about the rate of change of the quantity?

Solution

The function given is:

f(t)=2100(0.25)t12f(t) = 2100 \left(0.25\right)^{\frac{t}{12}}

Where f(t)f(t) represents the change in a quantity over tt months. The question asks about the constant 0.25 and its role in determining the rate of change of the quantity.

The constant 0.25 in the function represents the decay factor, indicating that the quantity is decreasing over time. Since 0.25 is less than 1, this suggests exponential decay, meaning that the quantity is shrinking as time progresses. More specifically, after each 12-month period, the quantity becomes 25% of its previous value.

To interpret the rate of change, we need to examine the decay rate. The exponent t12\frac{t}{12} shows that every 12 months, the quantity decreases to 25% of its previous value. In other words, the quantity reduces by a factor of 0.25 every year.

Would you like a more detailed breakdown of this, or do you have any other questions?

Here are some related questions for further exploration:

  1. How can you calculate the quantity after a specific number of months using the given function?
  2. How does exponential decay differ from linear decay?
  3. What is the value of f(t)f(t) after 24 months?
  4. How would you rewrite the function to model exponential growth instead of decay?
  5. How can the decay factor 0.250.25 be interpreted in terms of percentage?

Tip: To understand exponential decay more clearly, try plotting the function to visualize how the quantity decreases over time!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Exponential Decay
Rate of Change

Formulas

f(t) = 2100(0.25)^(t/12)

Theorems

Exponential Decay Rule

Suitable Grade Level

Grades 9-12