Question

n April of 2009, Mexico City experienced an outbreak of the H1N1 flu. If the spread of the disease can be modeled by y= 8,800,000/ 1+4189e^−0.2t , where t is the number of days after April 6th, on approximately what date did the total number of cases reach 100,000? Setup an equation and solve.

Answer 1

To find when H1N1 cases reached 100,000, we solve the exponential growth model equation by setting y to 100,000 and calculating for t. Solving this mathematically gives t ≈ 22.76 days after April 6th, which is roughly April 29th, 2009.

Step-by-step explanation:

The student is asking for help with a mathematics problem that involves solving for a variable in an exponential growth model. The model describes the spread of the H1N1 flu in Mexico City in April 2009. We are given the function y = 8,800,000 / (1 + 4189e−0.2t), and we need to determine the value of t, which represents the number of days after April 6th, when the total number of cases reached 100,000.

To solve for t, we set y to 100,000 and solve the equation:

1. 100,000 = 8,800,000 / (1 + 4189e−0.2t)
2. Multiply both sides by (1 + 4189e−0.2t) to clear the fraction: 100,000(1 + 4189e−0.2t) = 8,800,000
3. Divide both sides by 100,000: 1 + 4189e−0.2t = 88
4. Subtract 1 from both sides: 4189e−0.2t = 87
5. Divide both sides by 4189: e−0.2t ≈ 0.020777
6. Take the natural logarithm of both sides: −0.2t = ln(0.020777)
7. Solve for t: t = −(ln(0.020777))/0.2

Plugging the values into a calculator gives us approximately t = 22.76. Since we want the date when the cases reached 100,000, we add approximately 23 days to April 6th, which brings us to April 29th, 2009 as the approximate date on which the number of H1N1 flu cases in Mexico City reached 100,000.