Question

the equation \\(y = 4 \cdot 6^t\\) shows the number of infected people from an outbreak of the measles. the variable \\(y\\) represents the number of infected people, and \\(t\\) represents time in weeks.

in how many weeks will the number of infected people reach 568?

a.) 2.77 weeks
b.) 1.89 weeks
c.) 3.34 weeks
d.) 3.21 weeks

Explanation:

Set up the equation

We are given the exponential model \(y = 4 \cdot 6^t\), where \(y\) represents the number of infected people and \(t\) represents the time in weeks. We need to find the time \(t\) when the number of infected people \(y\) reaches \(568\).

Using the Exponential Equations knowledge point, we substitute \(y = 568\) into the equation:

$$568 = 4 \cdot 6^t$$

Isolate the exponential term

Using the Exponential Equations knowledge point

$$
LATEXBLOCK0
$$

Apply logarithms to solve for t

Using the Logarithmic Equations and Logarithm Definition knowledge points

$$
LATEXBLOCK1
$$

Calculate the numerical value

Using the Logarithm Properties knowledge point

$$
LATEXBLOCK2
$$

Answer:

• a.) 2.77 weeks (Correct answer)
• b.) 1.89 weeks
• c.) 3.34 weeks
• d.) 3.21 weeks