**Bacterial Growth Study**A strain of bacteria is placed into a nutrient broth at 30°C and allowed to grow. The data collected is presented in the table below. The population is measured in grams and the time in hours. Since the population \( P \) depends on time \( t \), and each input corresponds to exactly one output, we say that population is a function of time; so \( P(t) \) represents the population at time \( t \).**Table: Bacterial Growth Over Time**| Time (hr), \( t \) | Population (g), \( P \) ||-------------------|-------------------------|| 0 | 0.02 || 2.5 | 0.18 || 3.5 | 0.23 || 4.5 | 0.35 || 6 | 0.51 |---**Problem:**(a) Find the average rate of change of the population from 0 to 2.5 hours.To calculate the average rate of increase, use the formula:\[\text{Average Rate of Change} = \frac{P(t_2) - P(t_1)}{t_2 - t_1}\]where \( t_1 = 0 \), \( P(t_1) = 0.02 \), \( t_2 = 2.5 \), and \( P(t_2) = 0.18 \).On average, the population is increasing at a rate of \( \_\_\_ \) grams per hour from 0 to 2.5 hours. (Round to 3 decimal places as needed.)

A strain of bacteria is placed into a nutrient broth at 30° Time (hr), t Population (g), P C and allowed to grow. The data shown to the right are collected. The population is measured in grams and the time in hours. Since population P depends on time t and each input corresponds to exactly one output, we say that population is a function of time; so P(t) represents the population at time t. 0 NM 50 50 2.5 3.5 4.5 6 (a) Find the average rate of change of the population from 0 to 2.5 hours. On average, the population is increasing at a rate of (Round to 3 decimal places as needed.) 0.02 0.18 0.23 0.35 0.51 gram per hour from 0 to 2.5 hours.

A strain of bacteria is placed into a nutrient broth at 30° Time (hr), t Population (g), P C and allowed to grow. The data shown to the right are collected. The population is measured in grams and the time in hours. Since population P depends on time t and each input corresponds to exactly one output, we say that population is a function of time; so P(t) represents the population at time t. 0 NM 50 50 2.5 3.5 4.5 6 (a) Find the average rate of change of the population from 0 to 2.5 hours. On average, the population is increasing at a rate of (Round to 3 decimal places as needed.) 0.02 0.18 0.23 0.35 0.51 gram per hour from 0 to 2.5 hours.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Bacterial Growth Study**

A strain of bacteria is placed into a nutrient broth at 30°C and allowed to grow. The data collected is presented in the table below. The population is measured in grams and the time in hours. Since the population \( P \) depends on time \( t \), and each input corresponds to exactly one output, we say that population is a function of time; so \( P(t) \) represents the population at time \( t \).

**Table: Bacterial Growth Over Time**

| Time (hr), \( t \) | Population (g), \( P \) |
|-------------------|-------------------------|
| 0 | 0.02 |
| 2.5 | 0.18 |
| 3.5 | 0.23 |
| 4.5 | 0.35 |
| 6 | 0.51 |

---

**Problem:**

(a) Find the average rate of change of the population from 0 to 2.5 hours.

To calculate the average rate of increase, use the formula:

\[
\text{Average Rate of Change} = \frac{P(t_2) - P(t_1)}{t_2 - t_1}
\]

where \( t_1 = 0 \), \( P(t_1) = 0.02 \), \( t_2 = 2.5 \), and \( P(t_2) = 0.18 \).

On average, the population is increasing at a rate of \( \_\_\_ \) grams per hour from 0 to 2.5 hours. (Round to 3 decimal places as needed.)

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Follow-up Question

A strain of bacteria is placed into a nutrient broth at 30°C and allowed to grow. The data shown to the right are collected. The population is measured in grams and the time in hours. Since population \( P \) depends on time \( t \) and each input corresponds to exactly one output, we say that population is a function of time; so \( P(t) \) represents the population at time \( t \).

**Table**

- **Time (hr)**, \( t \)
- **Population (g)**, \( P \)

| Time (hr), \( t \) | Population (g), \( P \) |
|------------------|-----------------------|
| 0 | 0.02 |
| 2.5 | 0.18 |
| 3.5 | 0.23 |
| 4.5 | 0.35 |
| 6 | 0.51 |

**Questions:**

(a) Find the average rate of change of the population from 0 to 2.5 hours.

- On average, the population is increasing at a rate of \( 0.064 \) gram per hour from 0 to 2.5 hours. (Round to 3 decimal places as needed.)

(b) Find the average rate of change of the population from 4.5 to 6 hours.

- On average, the population is increasing at a rate of \(\_\_\) gram per hour from 4.5 to 6 hours. (Round to 3 decimal places as needed.)

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