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Key Concept

Exponential Growth and Decay

High SchoolMathematicsAlgebra

Exponential growth and decay models are used to describe phenomena that increase or decrease over time at a rate proportional to their current value. This concept is often applied in various fields including biology, economics, and physics, to understand the behavior of populations, investments, and radioactive decay, among others.

Basic Answer

Solution By Steps

Step 1: Understanding the Growth Rate The bacteria population increases by 20% every 10 minutes. This means if the initial population is $P$ , after 10 minutes it becomes $P + 0.2 P = 1.2 P$ .

Step 2: Setting Up the Equation for Doubling To find the time when the population doubles, we need to find $t$ such that the population becomes $2 P$ . We use the exponential growth formula: $P (t) = P_{0} \times (1.2)^{\frac{t}{10}}$ .

Step 3: Solving for $t$ We set $P (t) = 2 P$ and solve for $t$ : $2 P = P \times (1.2)^{\frac{t}{10}}$ $2 = (1.2)^{\frac{t}{10}}$

Step 4: Taking the Logarithm To solve for $t$ , take the natural logarithm on both sides: $ln (2) = \frac{t}{10} ln (1.2)$ $t = 10 \times \frac{ln ( 2 )}{ln ( 1.2 )}$

Step 5: Calculating the Value Using approximate values $ln (2) \approx 0.693$ and $ln (1.2) \approx 0.182$ : $t \approx 10 \times \frac{0.693}{0.182} \approx 38.1$

Final Answer

The time required for the bacteria population to double is approximately 38.1 minutes. The closest option is (b) 40 minutes.

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Super Answer

Solution By Steps

Step 1: Identify Growth Rate
The bacteria population increases at 20% every 10 minutes. This means if the initial population is P, after 10 minutes it becomes P * (1 + 0.20) = P * 1.2.

Step 2: Establish Doubling Condition
We need to find when the population becomes 2P (double the initial population).

Step 3: Calculate Time for Doubling
Let the number of 10-minute intervals required to double be n. The formula becomes:
P * (1.2)^n = 2P.

Step 4: Simplify the Equation
Cancelling P gives us:
(1.2)^n = 2.

Step 5: Apply Logarithms
Taking the logarithm of both sides:
n * log(1.2) = log(2).

Step 6: Solve for n
Thus,
n = log(2) / log(1.2).

Step 7: Calculate n Using Approximate Log Values
Using approximate values:
log(2) ≈ 0.301 and log(1.2) ≈ 0.0792,
n ≈ 0.301 / 0.0792 ≈ 3.8 (rounded to one decimal).

Step 8: Convert Intervals to Minutes
Since each interval is 10 minutes, total time = 3.8 * 10 = 38 minutes.

Step 9: Round to Nearest Answer Option
The closest answer choice to 38 minutes is 40 minutes.

Final Answer

40 minutes

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