Write an exponential equation in the form y = a(b)^x that can model the number of bacteria, y, after x hours in sunlight.

Understand the Problem

The question is asking us to formulate an exponential equation to model the growth of bacteria based on initial and observed quantities over a period of time. We will determine the parameters 'a' and 'b' based on the counts of bacteria before and after one hour of exposure to sunlight.

Answer

The exponential equation is $y = 5(7)^x$.

Steps to Solve

Identify the Initial Values

The equation we will use is in the form $y = a(b)^x$, where:

$y$ is the number of bacteria after $x$ hours
$a$ is the initial amount of bacteria
$b$ is the growth factor
$x$ is the time in hours

From the problem, we know that at $x = 0$ (initial time), $y = 5$. Hence, we have: $$ a = 5 $$

Set Up the Equation After One Hour

After one hour ($x = 1$), the number of bacteria increases to 35. We can set up the equation using our known values: $$ 35 = 5(b)^1 $$

Solve for the Growth Factor $b$

Now, we need to solve for $b$ in the equation: $$ 35 = 5b $$

Dividing both sides by 5, we get: $$ b = \frac{35}{5} = 7 $$

Write the Final Exponential Equation

Substituting the values of $a$ and $b$ back into the original equation, we get: $$ y = 5(7)^x $$

The exponential equation that models the number of bacteria after $x$ hours in sunlight is: $$ y = 5(7)^x $$

More Information

This equation shows that the initial bacteria count starts at 5, and every hour, the number of bacteria multiplies by 7. This illustrates rapid bacterial growth, which is common in favorable conditions such as sunlight.

Tips

Confusing the Initial Count with Growth Factor: It's important to ensure that the initial count of 5 bacteria is set as the value of $a$, not $b$.
Forgetting to Solve for $b$: Make sure to isolate $b$ correctly from the equation.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!