Math Problem Statement

Round to three decimal places

Evaluate f(x) = 250e^{0.05x} at x = 20, round to three decimal places.

Solution

The given function is:

f(x)=250e0.05xf(x) = 250 e^{0.05x}

You are asked to evaluate this at x=20x = 20 and round the result to three decimal places.

Let's calculate f(20)f(20):

f(20)=250e0.05×20f(20) = 250 e^{0.05 \times 20}

First, calculate 0.05×20=10.05 \times 20 = 1, so:

f(20)=250e1f(20) = 250 e^1

Since e2.71828e \approx 2.71828:

f(20)=250×2.71828679.57f(20) = 250 \times 2.71828 \approx 679.57

Thus, f(20)679.570f(20) \approx 679.570 when rounded to three decimal places.

Let me know if you want more details or have any other questions!

Here are 5 follow-up questions to consider:

  1. How does the exponential function behave for other values of xx?
  2. What is the significance of the constant ee in exponential growth?
  3. How would the graph of this function look for values of xx between 0 and 20?
  4. What happens to f(x)f(x) as xx approaches infinity?
  5. How would changing the coefficient 0.050.05 affect the growth of the function?

Tip: When dealing with exponential functions, the base ee represents natural growth processes and has applications in fields like biology, economics, and physics!

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Math Problem Analysis

Mathematical Concepts

Exponential Growth
Natural Exponential Function
Algebra

Formulas

f(x) = a e^{bx}

Theorems

Exponential Growth

Suitable Grade Level

Grades 10-12