Answered

Insect larvae grow incredibly quickly over the first several days of their lives. The length of one particular species increases exponentially at a rate of $25 \%$ each day. A scientist studying this species of larva measures one specimen to be 3 mm in length at the beginning of his observation period. The scientist will transfer the larvae to a new environment once the larvae are over 9.15 millimeters long.



If $t$ represents the number of days since the scientist began his observations, write an inequality to represent the situation, and use it to determine after how many days the larvae are transferred to the new environment.


A$t < 6$
B$t < 3$
C$t > 5$
D$t > 9$

Verified Answer :

  • Set up the exponential growth model \(L(t) = 3(1.25)^t\).
  • Establish the transfer inequality: \(3(1.25)^t > 9.15\), which simplifies to \((1.25)^t > 3.05\).
  • Test integer values to find that the condition is met when \(t > 5\).
  • Select the matching option: \(\boxed{t > 5}\).

Explanation

  1. Formulating the Growth Model

The growth of the larvae can be modeled using the formula for exponential growth: \(L(t) = L_0(1 + r)^t\), where \(L_0\) is the initial length, \(r\) is the daily growth rate, and \(t\) is the number of days.
Given that the initial length \(L_0 = 3 \text{ mm}\) and the growth rate \(r = 25\% = 0.25\), the model becomes:
\[L(t) = 3(1 + 0.25)^t = 3(1.25)^t\]
The scientist will transfer the larvae when the length exceeds \(9.15 \text{ mm}\). Therefore, we need to solve the inequality:
\[3(1.25)^t > 9.15\]

  1. Solving the Inequality

To solve the inequality \(3(1.25)^t > 9.15\), we first isolate the exponential term by dividing both sides by \(3\):
\[(1.25)^t > \frac{9.15}{3}\]
\[(1.25)^t > 3.05\]
We need to find the value of \(t\) for which this inequality holds. Let's test integer values for \(t\):

  • For \(t = 5\): \((1.25)^5 \approx 3.0517578\). Since \(3.0517578 > 3.05\), the condition is satisfied at \(t = 5\).
  • For values of \(t\) larger than \(5\), the value of \((1.25)^t\) will continue to increase.

Thus, the larvae will be transferred when \(t > 5\).

  1. Conclusion and Final Answer

Comparing our result \(t > 5\) with the given options, we see that it corresponds to one of the choices provided. Therefore, the correct choice is \(t > 5\). This confirms that after 5 days, the larvae's length surpasses the threshold of \(9.15 \text{ mm}\), necessitating their transfer. The final answer is \(\boxed{t > 5}\).

Examples

Exponential growth is a common concept in biology, used to predict the population dynamics of bacteria, the spread of viruses, or even the growth of cells in a lab. For example, if a researcher knows that a specific culture of bacteria doubles every hour, they can use an exponential growth model to determine exactly when the population will reach a size that would require a larger containment vessel. Understanding these models helps in planning experiments and ensuring that resources are managed effectively.

Related Questions