Chapter 10: Q. 10.87 (page 1052)

Researchers recorded that a certain bacteria population grew from $100$ to $500$ in $6$ hours. At this rate of growth,
how many bacteria will there be $24$ hours from the start of the experiment?

Short Answer

Expert verified

The number of bacteria will be there in $24$ hours is $62142$ .

Step by step solution

Step 1. Given information

It is given that bacteria grew from $100$ to $500$ in $6$ hours.

Step 2. Finding the value of k

For an original amount, $A_{0}$ , that grows or decays at a rate, $k$ , for a certain time, $t$ , the final amount, $A$ , is:

$A = A_{0} e^{k t}$

Then substituting the given values in the formula we get:

$A = A_{0} e^{k t} 500 = 100 e^{6 k} 5 = e^{6 k}$

Step 3. Use logarithmic properties

Taking logarithmic properties on both sides, we get:

$5 = e^{6 k} \ln 5 = \ln e^{6 k} \ln 5 = 6 k [\ln e^{a} = a] k = \frac{\ln 5}{6} k = \frac{1.609}{6} k \approx 0.268$

Step 4. Finding the number of increased bacteria

Substituting the obtained value of kand the given time in the given formula, we get:

$A = A_{0} e^{k t} A = 100 e^{0.268 \cdot 24} A = 100 e^{6.432} A = 100 \cdot 621.41 A = 62141.553 A \approx 62142$

Thus, $62142$ bacteria will be there after $24$ hours.

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Researchers recorded that a certain bacteria population grew from $100$ to $500$ in $6$ hours. At this rate of growth,
how many bacteria will there be $24$ hours from the start of the experiment?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of k

Step 3. Use logarithmic properties

Step 4. Finding the number of increased bacteria

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Most popular questions from this chapter

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Researchers recorded that a certain bacteria population grew from 100 to 500in 6 hours. At this rate of growth,how many bacteria will there be 24 hours from the start of the experiment?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of k

Step 3. Use logarithmic properties

Step 4. Finding the number of increased bacteria

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Researchers recorded that a certain bacteria population grew from $100$ to $500$ in $6$ hours. At this rate of growth,
how many bacteria will there be $24$ hours from the start of the experiment?