1:52 Done nonsmokers? Justify your answer. Select an answer Select an answer Submit Part A scientist finds that an intial bacteria count of 5 million grows to 75 million after 3 days. Write an exponential equation of the form \( y=a b^{x} \) that models this scenario. Round \( b \) to four decimal places. \( y=\square \) Question Help: 0 Details Submit Question

To model the growth of the bacteria, we start with the equation \( y = a b^x \). Here, \( a \) is the initial amount of bacteria, so \( a = 5,000,000 \). After 3 days (where \( x = 3 \)), the bacteria count is \( y = 75,000,000 \). Substituting these values into the equation gives us: \[ 75,000,000 = 5,000,000 \times b^3 \] By simplifying this, we get: \[ 15 = b^3 \] Now, taking the cube root of both sides: \[ b \approx 2.466 \] Rounding to four decimal places, we find \( b \approx 2.4660 \). Now we've modeled our equation as: \[ y = 5,000,000 (2.4660)^x \] This equation estimates bacteria growth over time in an exponential manner. In many real-world applications, especially in ecology and microbiology, understanding bacterial growth is essential. For instance, knowing how bacteria multiply can help us in managing infections or in biotechnological processes where bacteria are used to produce biofuels or pharmaceuticals. When working with exponential growth equations, it’s important to avoid common pitfalls. One mistake is neglecting to ensure that all units consistently reflect the same time frame, like hours or days. Additionally, remember to check your growth factor \( b \)—if it’s too high or low, your estimates can deviate dramatically from reality!