Question

Using the model P(t) = P0 * e^(-kt), determine how long it will take for 75% of the population of a country to be infected with a virus with an infection rate of 9.8%. Round the answer to the nearest year.

Answer 1

Step-by-step explanation:

To determine how long it will take for 75% of the population of a country to be infected with a virus, we can use the formula P(t) = P0 * e^(-kt), where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the infection rate, and t is the time in years.

In this case, we want to find the value of t when P(t) is equal to 75% of the initial population (P0). Let's assume the initial population is denoted as P0.

So we have the equation 75% of P0 = P0 * e^(-kt).

To solve for t, we need to isolate it on one side of the equation.

Dividing both sides of the equation by P0 gives us: 0.75 = e^(-kt).

To isolate the exponential term, we can take the natural logarithm (ln) of both sides of the equation.

ln(0.75) = ln(e^(-kt)).

Using the property of logarithms, we can bring down the exponent: ln(0.75) = -kt * ln(e).

Since ln(e) is equal to 1, the equation simplifies to: ln(0.75) = -kt.

Now, we can solve for t by dividing both sides of the equation by -k: t = ln(0.75) / -k.

Using the given infection rate of 9.8% (or 0.098), we can substitute it into the equation to find the value of t.

t = ln(0.75) / -0.098.

Calculating this value, we find t ≈ 7.06 years.

Rounding to the nearest year, it will take approximately 7 years for 75% of the population to be infected with the virus

Answer 2

The solution involves setting up the exponential decay model to represent 75% population infection, solving for time when the infection rate is 9.8%, and rounding the result to the nearest year.

Step-by-step explanation:

The student's question involves using the exponential decay model P(t) = P0 * e^(-kt) to find how long it will take for 75% of a population to be infected with a virus at a given infection rate.

To solve this, we set up the equation to reflect the condition that P(t) is equal to 75% of P0, and then solve for t when the infection rate k is 9.8%, which is 0.098 when expressed as a decimal. The equation becomes P0 * 0.75 = P0 * e^(-0.098t), which simplifies to 0.75 = e^(-0.098t).

Taking the natural logarithm of both sides of the equation gives us ln(0.75) = -0.098t. Solving for t, we find t = ln(0.75) / -0.098, which can be calculated using a scientific calculator. Once we have the value for t, we round it to the nearest year to obtain the student's answer.