Math Problem Statement

The exponential function f(x)=abx models the volume of water in an inverted cone after x seconds of draining. As x increases, the instantaneous rate of change gets closer to 0.

What is a possible growth factor for f(x) ?

One possible growth factor is 1.

One possible growth factor is –1.56.

One possible growth factor is 0.33.

One possible growth factor is 1.36.

Solution

For the given exponential function f(x)=abxf(x) = ab^x, where aa is a constant and bb is the base (growth factor), the behavior of the function depends on the value of bb.

Given that the function models the volume of water draining from an inverted cone, and the rate of change approaches 0 as xx increases, this suggests that the volume is decreasing over time.

Now, let's examine the possible growth factors:

  • 1: A growth factor of 1 means no change in the volume over time, which doesn't fit the description of water draining from the cone.
  • –1.56: A negative growth factor is unrealistic in this context because it would imply alternating values (positive and negative), which is not possible for volume.
  • 0.33: A growth factor between 0 and 1 indicates exponential decay, which aligns with the scenario of water draining, as the volume decreases over time.
  • 1.36: A growth factor greater than 1 would imply exponential growth, which is not appropriate for this situation since the water volume is decreasing.

Thus, the possible growth factor is 0.33 because it represents exponential decay, fitting the situation of water draining from the cone.

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

  1. How does the initial value aa influence the function f(x)f(x)?
  2. What is the meaning of the growth factor in the context of exponential functions?
  3. How would the function behave if the growth factor was exactly 1?
  4. What happens to the function f(x)=abxf(x) = ab^x when b>1b > 1?
  5. Can the growth factor in an exponential function be negative? Why or why not?

Tip: Exponential decay occurs when the growth factor bb is between 0 and 1, which results in the function decreasing over time.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Decay Functions
Rate of Change

Formulas

Exponential function formula f(x) = ab^x

Theorems

Exponential Growth and Decay Theorem

Suitable Grade Level

Grades 9-12