Chapter 10: Q. 10.89 (page 1053)

The half-life of magnesium- $27$ is $9.45$ minutes. How much of a $10$ -mg sample will be left in $6$ minutes?

Short Answer

Expert verified

$6.45 m g$ of the sample will be left from $10$ mg in $6$ minutes

Step by step solution

Step 1. Given information

It is given that the half-life of the element is $9.45$ minutes.

Step 2. Finding the value of k

For an original amount, $A_{0}$ , that grows or decays at a rate, localid="1645080842678" $k$ , for a certain time, $t$ , the final amount,

Then substituting the given values in the formula we get:

localid="1645079763896" $A = A_{0} e^{k t} 5 = 10 e^{9.45 k} \frac{1}{2} = e^{9.45 k}$

Step 3. Use logarithmic properties

Taking logarithmic properties on both sides, we get:

$\frac{1}{2} = e^{9.45 k} \ln \frac{1}{2} = \ln e^{9.45 k} \ln \frac{1}{2} = 9.45 k [\ln e^{a} = a] \frac{- 0.69}{9.45} = k k = - 0.073$

Step 4. Finding the number of decreased bacteria

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

$A = A_{0} e^{k t} A = 10 e^{- 0.073 \cdot 6} A = 10 e^{- 0.438} A = 10 \cdot 0.645 A = 6.45$

role="math" localid="1645080094787" $6.45 m g$ of the sample will be left in $6$ minutes.

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The half-life of magnesium- $27$ is $9.45$ minutes. How much of a $10$ -mg sample will be left in $6$ minutes?

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 decreased bacteria

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The half-life of magnesium-27 is 9.45 minutes. How much of a 10-mg sample will be left in 6 minutes?

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 decreased bacteria

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Discrete Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

The half-life of magnesium- $27$ is $9.45$ minutes. How much of a $10$ -mg sample will be left in $6$ minutes?