To find the area between the birth rate curve \( b(t) \) and the death rate curve \( d(t) \) over the interval \( 0 \leq t \leq 10 \), we first need to set up the integral of the difference between the two functions over this interval.
The functions are given as:
- Birth rate:
\[ b(t) = 2500 e^{0.024t} \] - Death rate:
\[ d(t) = 1400 e^{0.018t} \]
The area \( A \) between the two curves from \( t = 0 \) to \( t = 10 \) is given by the integral of their difference: \[ A = \int_0^{10} (b(t) - d(t)) , dt = \int_0^{10} (2500 e^{0.024t} - 1400 e^{0.018t}) , dt \]
We will calculate this integral step by step.
Step 1: Calculate the integral
Compute the integral: \[ A = \int_0^{10} (2500 e^{0.024t} - 1400 e^{0.018t}) , dt \]
This can be split into two separate integrals: \[ A = \int_0^{10} 2500 e^{0.024t} , dt - \int_0^{10} 1400 e^{0.018t} , dt \]
Integrating \( 2500 e^{0.024t} \)
The integral of \( e^{kt} \) is \( \frac{1}{k} e^{kt} \). Thus, we have: \[ \int 2500 e^{0.024t} , dt = \frac{2500}{0.024} e^{0.024t} = \frac{2500}{0.024} e^{0.024t} \]
Evaluate from 0 to 10: \[ = \left. \frac{2500}{0.024} e^{0.024t} \right|_0^{10} = \frac{2500}{0.024} \left( e^{0.24} - 1 \right) \]
Integrating \( 1400 e^{0.018t} \)
Similarly, \[ \int 1400 e^{0.018t} , dt = \frac{1400}{0.018} e^{0.018t} \]
Evaluate from 0 to 10: \[ = \left. \frac{1400}{0.018} e^{0.018t} \right|_0^{10} = \frac{1400}{0.018} \left( e^{0.18} - 1 \right) \]
Step 2: Putting it all together
Now substitute back into the area expression: \[ A = \frac{2500}{0.024} \left(e^{0.24} - 1\right) - \frac{1400}{0.018} \left(e^{0.18} - 1\right) \]
Step 3: Calculate numerical values
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Calculate \( e^{0.24} \) and \( e^{0.18} \):
- \( e^{0.24} \approx 1.271 \)
- \( e^{0.18} \approx 1.197 \)
-
Compute the area: \[ A \approx \frac{2500}{0.024}(1.271 - 1) - \frac{1400}{0.018}(1.197 - 1) \] \[ \approx \frac{2500}{0.024} \cdot 0.271 - \frac{1400}{0.018} \cdot 0.197 \] \[ \approx 104166.67 \cdot 0.271 - 77777.78 \cdot 0.197 \] \[ \approx 28219.44 - 15370.50 \approx 12848.94 \]
Result
Rounding to the nearest integer, the area between the curves over the interval \( 0 \leq t \leq 10 \) is approximately: \[ \boxed{12849} \]