The concentration of benzene is 490 PPM 3 hours after a spill. 8 hours after the spill the concentration has fallen to 325 PPM. Find the Continuous Exponential model

To find the continuous exponential model for the concentration of benzene over time, we assume the concentration \( C(t) \) follows the form:

\[ C(t) = C_0 e^{-kt} \]

where:
- \( C(t) \) is the concentration at time \( t \),
- \( C_0 \) is the initial concentration at \( t = 0 \),
- \( k \) is the decay constant.

We are given:
- \( C(3) = 490 \) PPM (3 hours after the spill),
- \( C(8) = 325 \) PPM (8 hours after the spill).

Let's set up two equations using these data points:

1. \( 490 = C_0 e^{-3k} \)
2. \( 325 = C_0 e^{-8k} \)

To find \( C_0 \) and \( k \), we can follow these steps:

### Step 1: Solve for \( k \)
Divide the second equation by the first equation to eliminate \( C_0 \):

\[
\frac{325}{490} = \frac{C_0 e^{-8k}}{C_0 e^{-3k}} \implies \frac{325}{490} = e^{-8k + 3k} \implies \frac{325}{490} = e^{-5k}
\]

Taking the natural logarithm of both sides:

\[
\ln\left(\frac{325}{490}\right) = -5k
\]

\[
k = -\frac{1}{5} \ln\left(\frac{325}{490}\right)
\]

Calculate \( k \):

\[
k = -\frac{1}{5} \ln\left(\frac{325}{490}\right) \approx -\frac{1}{5} \ln(0.6633) \approx -\frac{1}{5} \times (-0.4105) \approx 0.0821
\]

### Step 2: Solve for \( C_0 \)
Use one of the original equations to solve for \( C_0 \). Let's use the first equation:

\[
490 = C_0 e^{-3k}
\]

Substitute \( k \):

\[
490 = C_0 e^{-3 \times 0.0821} \implies 490 = C_0 e^{-0.2463}
\]

\[
C_0 = 490 / e^{-0.2463} \approx 490 / 0.7818 \approx 626.83
\]

### Continuous Exponential Model
Now we have both \( C_0 \) and \( k \). The continuous exponential model is:

\[
C(t) = 626.83 e^{-0.0821t}
\]

Would you like more details or have any questions?

Here are 5 more questions you might find interesting:
1. How do you solve a system of linear equations?
2. What is the derivative of a polynomial function?
3. How do you find the integral of \( e^{ax} \)?
4. What is the quadratic formula and how is it used?
5. Can you explain the concept of limits in calculus?

**Tip:** When solving exponential models, always check your units and make sure the base and exponent align with the context of your problem.

Explore how to calculate the continuous exponential model for benzene concentration decay over time. This problem involves finding the decay constant and initial concentration using given data points. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation