The equation y equals 25 left parenthesis 2 to the power of t right parenthesis shows the number of infected people from an outbreak of E.coli. 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 800?


We are given the equation:

\[
y = 25 \cdot 2^t
\]

where \( y \) represents the number of infected people and \( t \) represents the time in weeks. We need to find the value of \( t \) when \( y = 800 \).

### Step 1: Set up the equation
Substitute \( y = 800 \) into the equation:

\[
800 = 25 \cdot 2^t
\]

### Step 2: Isolate the exponential term
To solve for \( t \), first isolate \( 2^t \):

\[
\frac{800}{25} = 2^t
\]

Simplify the left side:

\[
32 = 2^t
\]

### Step 3: Express \( 32 \) as a power of 2
We know that \( 32 = 2^5 \), so:

\[
2^5 = 2^t
\]

### Step 4: Equate the exponents
Since the bases are the same, the exponents must be equal:

\[
t = 5
\]

Thus, it will take \(\boxed{5}\) weeks for the number of infected people to reach 800.

Would you like any further details or have any additional questions on this solution?

### Relative Questions
1. How would the solution change if the initial number of infected people was different?
2. Can you explain the process of solving exponential equations?
3. What does the base of 2 represent in this context?
4. How would you solve the problem if the equation had a different base, say 3, instead of 2?
5. How would the growth rate change if the exponent was halved?

### Tip
When dealing with exponential growth, always check if the numbers can be expressed as powers of the base to simplify the equation quickly.

Learn how to solve the exponential growth equation y = 25 * 2^t to determine how many weeks it takes for the number of infected people to reach 800. This problem involves understanding exponential growth, isolating exponential terms, and solving equations. Suitable for high school students.

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