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Question 17 (5 points)

Among various populations of plants or animals, diseases spread exponentially. Use the function [tex]$y=8000\left(1-e^{-0.03 t}\right)[tex]$[/tex] to model the spread of Common Corn Rust through a field of 8000 corn plants, with [tex]$[/tex]t$[/tex] equal to the number of days since the first case of the disease.

How many plants will be infected with Common Corn Rust after 10 days?

A. 2073
B. 236
C. 761
D. 5927


Sagot :

To determine how many corn plants will be infected with Common Corn Rust after 10 days, we will use the given exponential model \( y = 8000 \left( 1 - e^{-0.03 t} \right) \). Here, \( y \) represents the number of infected plants, and \( t \) represents the number of days since the first case of the disease.

We are tasked with finding the number of infected plants after 10 days, so we will substitute \( t = 10 \) into the model.

1. Substitute \( t = 10 \) into the equation:
[tex]\[ y = 8000 \left( 1 - e^{-0.03 \times 10} \right) \][/tex]

2. Simplify the exponent in the exponential term:
[tex]\[ y = 8000 \left( 1 - e^{-0.3} \right) \][/tex]

3. Compute \( e^{-0.3} \) using the exponential constant \( e \):
[tex]\[ e^{-0.3} \approx 0.7408182207 \][/tex]

4. Substitute the value back into the equation:
[tex]\[ y = 8000 \left( 1 - 0.7408182207 \right) \][/tex]

5. Subtract \( 0.7408182207 \) from 1:
[tex]\[ y = 8000 \left( 0.2591817793 \right) \][/tex]

6. Multiply the constants:
[tex]\[ y \approx 2073.454234546257 \][/tex]

After completing the calculations, we find that approximately 2073 plants will be infected with Common Corn Rust after 10 days. Thus, the correct answer is 2073.
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