Math Problem Statement

Find the exponential function y=y0e^(kt), with points (3,5) and (0,2)

Solution

To find the exponential function of the form y=y0ekty = y_0 e^{kt} using the points (3,5)(3,5) and (0,2)(0,2), follow these steps:

Step 1: Use the point (0,2)(0, 2)

From the equation y=y0ekty = y_0 e^{kt}, when t=0t = 0, the equation becomes:

y0=y0ek(0)=y01=2y_0 = y_0 e^{k(0)} = y_0 \cdot 1 = 2

Thus, y0=2y_0 = 2.

Step 2: Use the point (3,5)(3, 5)

Now, substitute the values t=3t = 3 and y=5y = 5 into the equation y=y0ekty = y_0 e^{kt}, where y0=2y_0 = 2:

5=2e3k5 = 2 e^{3k}

Solve for e3ke^{3k}:

52=e3k\frac{5}{2} = e^{3k}

e3k=2.5e^{3k} = 2.5

Step 3: Solve for kk

Take the natural logarithm of both sides to solve for kk:

ln(e3k)=ln(2.5)\ln(e^{3k}) = \ln(2.5)

3k=ln(2.5)3k = \ln(2.5)

k=ln(2.5)3k = \frac{\ln(2.5)}{3}

Using a calculator, we get:

k0.916330.3054k \approx \frac{0.9163}{3} \approx 0.3054

Final Answer:

The exponential function is:

y=2e0.3054ty = 2 e^{0.3054t}

If you'd like further details or have any questions, feel free to ask!

Here are some related questions you may find interesting:

  1. How does changing the value of y0y_0 affect the graph of the exponential function?
  2. What is the behavior of the function as tt \to \infty?
  3. How can we derive exponential growth or decay from the general equation?
  4. What happens if kk is negative in this exponential model?
  5. How would the function change if the point (0, 2) was replaced by (0, 5)?

Tip: To understand exponential functions better, plot them for different values of kk to see how the growth rate changes.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponential Functions
Logarithms
Algebra

Formulas

y = y0 e^(kt)
ln(x)

Theorems

Properties of Exponential Functions
Natural Logarithm Properties

Suitable Grade Level

Grades 10-12