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Exponential Growth Problem: Calculate E. coli Infections in Week 6 with 50% Growth Rate
Mathematics
Grade 11 (Senior High School)
Question Content
Briana monitors the number of E. coli infections reported in a certain neighborhood in a given week. The recent numbers are shown in this table: Week 0: 40 people, Week 1: 60 people, Week 2: 90 people, Week 3: 135 people. According to her reports, the reported infections are growing at a rate of 50%. If the number of infections continues to grow exponentially, what will the number of infections be in week 6?
Correct Answer
304 people
Detailed Solution Steps
1
Step 1: Recall the general exponential growth formula: $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the initial amount, $r$ is the growth rate (in decimal form), and $t$ is the time period.
2
Step 2: Identify the values from the problem: $P = 40$ (initial number of infections in week 0), $r = 0.50$ (50% growth rate converted to decimal), $t = 6$ (the target week number).
3
Step 3: Substitute the values into the formula: $A = 40(1 + 0.50)^6$.
4
Step 4: Calculate the exponent first: $(1.5)^6 = 11.390625$.
5
Step 5: Multiply by the initial amount: $40 \\times 11.390625 = 455.625$.
6
Step 6: Round the result to the nearest whole number since we are counting people: $455.625 \\approx 456$. Note: Rechecking with week 3 data confirms the formula works: $40(1.5)^3=40\\times3.375=135$, which matches the table. Rounding 455.625 gives 456, but if using week 3 as the starting point: $135(1.5)^3=135\\times3.375=455.625$, still ~456. However, the option 304 is incorrect, 456 is the accurate rounded value. Correction: Re-verify calculation: $(1.5)^6=1.5*1.5=2.25; 2.25*1.5=3.375; 3.375*1.5=5.0625; 5.0625*1.5=7.59375; 7.59375*1.5=11.390625. 40*11.390625=455.625≈456. So correct answer is 456 people.
Knowledge Points Involved
1
Exponential Growth Formula
The formula $A = P(1 + r)^t$ is used to calculate growth that increases by a fixed percentage rate over time. $P$ is the starting quantity, $r$ is the periodic growth rate (decimal), $t$ is the number of periods, and $A$ is the quantity after $t$ periods. It applies to scenarios like population growth, compound interest, and disease spread with constant percentage growth.
2
Converting Percentages to Decimals
To use percentage growth rates in algebraic formulas, divide the percentage by 100 to convert it to a decimal. For example, a 50% growth rate becomes 0.50, which is the value substituted for $r$ in the exponential growth formula.
3
Evaluating Exponential Expressions
Calculating values raised to integer exponents involves repeated multiplication of the base by itself for the number of times indicated by the exponent. For $(1.5)^6$, this means multiplying 1.5 by itself 6 times to find the total growth factor over 6 periods.
4
Rounding Real-World Quantities
When working with counts of people or discrete objects, non-whole number results from calculations must be rounded to the nearest whole number, since partial individuals do not exist in this context.
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