Math / Algebra

QuestionThe half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a mass of 4 mg . What was the initial mass (in mg ) of the sample? $128$
What is the mass 4 weeks after the start? 1

Studdy Solution

STEP 1

What is this asking? We're figuring out how much Palladium-100 we started with if we only have 4 mg left after 20 days, and then, how much will be left after 4 weeks! Watch out! Don't mix up days and weeks, and make sure to understand how half-life works!
Remember, half-life is the time it takes for half of the substance to decay.

STEP 2

1. Calculate the initial mass.
2. Calculate the mass after 4 weeks.

STEP 3

Alright, so the half-life is 4 days, and we're looking at a period of 20 days.
Let's see how many half-lives that is: $\frac{20 \text{ days}}{4 \text{ days/half-life}} = \textbf{5 half-lives}$ .

STEP 4

This means the sample has been halved 5 times.
After one half-life, we have $\frac{1}{2}$ left.
After two, we have $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ left.
Following this pattern, after five half-lives, we'll have $(\frac{1}{2})^5 = \frac{1}{32}$ of the initial mass remaining.

STEP 5

We know that after 20 days (5 half-lives), we have 4 mg left.
So, $\frac{1}{32}$ of the initial mass is equal to 4 mg.
Let's call the initial mass $m_0$ .
We can write this as an equation: $\frac{1}{32} \cdot m_0 = 4$ .

STEP 6

To find $m_0$ , we can multiply both sides of the equation by 32: $32 \cdot \frac{1}{32} \cdot m_0 = 4 \cdot 32$ .
This simplifies to $m_0 = \textbf{128 mg}$ .
So, the initial mass of the sample was 128 mg.

STEP 7

First, let's convert 4 weeks into days: $4 \text{ weeks} \cdot 7 \text{ days/week} = \textbf{28 days}$ .

STEP 8

Now, let's figure out how many half-lives are in 28 days: $\frac{28 \text{ days}}{4 \text{ days/half-life}} = \textbf{7 half-lives}$ .

STEP 9

After 7 half-lives, the remaining mass will be $(\frac{1}{2})^7 = \frac{1}{128}$ of the initial mass.

STEP 10

We found that the initial mass was 128 mg.
So, after 28 days (7 half-lives), the remaining mass will be $\frac{1}{128} \cdot 128 \text{ mg} = \textbf{1 mg}$ .

STEP 11

The initial mass of the Palladium-100 sample was 128 mg.
After 4 weeks, the remaining mass will be 1 mg.

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QuestionThe half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a mass of 4 mg . What was the initial mass (in mg ) of the sample? $128$
What is the mass 4 weeks after the start? 1

Studdy Solution

STEP 1

STEP 2

1. Calculate the initial mass.
2. Calculate the mass after 4 weeks.

STEP 3

Alright, so the half-life is 4 days, and we're looking at a period of 20 days.
Let's see how many half-lives that is: $\frac{20 \text{ days}}{4 \text{ days/half-life}} = \textbf{5 half-lives}$ .

STEP 4

This means the sample has been halved 5 times.
After one half-life, we have $\frac{1}{2}$ left.
After two, we have $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ left.
Following this pattern, after five half-lives, we'll have $(\frac{1}{2})^5 = \frac{1}{32}$ of the initial mass remaining.

STEP 5

We know that after 20 days (5 half-lives), we have 4 mg left.
So, $\frac{1}{32}$ of the initial mass is equal to 4 mg.
Let's call the initial mass $m_0$ .
We can write this as an equation: $\frac{1}{32} \cdot m_0 = 4$ .

STEP 6

To find $m_0$ , we can multiply both sides of the equation by 32: $32 \cdot \frac{1}{32} \cdot m_0 = 4 \cdot 32$ .
This simplifies to $m_0 = \textbf{128 mg}$ .
So, the initial mass of the sample was 128 mg.

STEP 7

First, let's convert 4 weeks into days: $4 \text{ weeks} \cdot 7 \text{ days/week} = \textbf{28 days}$ .

STEP 8

Now, let's figure out how many half-lives are in 28 days: $\frac{28 \text{ days}}{4 \text{ days/half-life}} = \textbf{7 half-lives}$ .

STEP 9

After 7 half-lives, the remaining mass will be $(\frac{1}{2})^7 = \frac{1}{128}$ of the initial mass.

STEP 10

We found that the initial mass was 128 mg.
So, after 28 days (7 half-lives), the remaining mass will be $\frac{1}{128} \cdot 128 \text{ mg} = \textbf{1 mg}$ .

STEP 11

The initial mass of the Palladium-100 sample was 128 mg.
After 4 weeks, the remaining mass will be 1 mg.

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QuestionThe half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a mass of 4 mg . What was the initial mass (in mg ) of the sample? 128128128What is the mass 4 weeks after the start? 1

Studdy Solution

STEP 1

STEP 2

1. Calculate the initial mass.2. Calculate the mass after 4 weeks.

STEP 3

Alright, so the half-life is **4 days**, and we're looking at a period of **20 days**.Let's see how many half-lives that is: 20 days4 days/half-life=5 half-lives\frac{20 \text{ days}}{4 \text{ days/half-life}} = \textbf{5 half-lives}4 days/half-life20 days​=5 half-lives.

STEP 4

STEP 5

We know that after 20 days (5 half-lives), we have **4 mg** left.So, 132\frac{1}{32}321​ of the **initial mass** is equal to 4 mg.Let's call the **initial mass** m0m_0m0​.We can write this as an equation: 132⋅m0=4\frac{1}{32} \cdot m_0 = 4321​⋅m0​=4.

STEP 6

To find m0m_0m0​, we can multiply both sides of the equation by 32: 32⋅132⋅m0=4⋅3232 \cdot \frac{1}{32} \cdot m_0 = 4 \cdot 3232⋅321​⋅m0​=4⋅32.This simplifies to m0=128 mgm_0 = \textbf{128 mg}m0​=128 mg.So, the **initial mass** of the sample was **128 mg**.

STEP 7

First, let's convert **4 weeks** into days: 4 weeks⋅7 days/week=28 days4 \text{ weeks} \cdot 7 \text{ days/week} = \textbf{28 days}4 weeks⋅7 days/week=28 days.

STEP 8

Now, let's figure out how many half-lives are in 28 days: 28 days4 days/half-life=7 half-lives\frac{28 \text{ days}}{4 \text{ days/half-life}} = \textbf{7 half-lives}4 days/half-life28 days​=7 half-lives.

STEP 9

After 7 half-lives, the remaining mass will be (12)7=1128(\frac{1}{2})^7 = \frac{1}{128}(21​)7=1281​ of the **initial mass**.

STEP 10

We found that the **initial mass** was **128 mg**.So, after 28 days (7 half-lives), the remaining mass will be 1128⋅128 mg=1 mg\frac{1}{128} \cdot 128 \text{ mg} = \textbf{1 mg}1281​⋅128 mg=1 mg.

STEP 11

The initial mass of the Palladium-100 sample was **128 mg**.After 4 weeks, the remaining mass will be **1 mg**.

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QuestionThe half-life of Palladium-100 is 4 days. After 20 days a sample of Palladium-100 has been reduced to a mass of 4 mg . What was the initial mass (in mg ) of the sample? $128$
What is the mass 4 weeks after the start? 1

1. Calculate the initial mass.
2. Calculate the mass after 4 weeks.

Alright, so the half-life is 4 days, and we're looking at a period of 20 days.
Let's see how many half-lives that is: $\frac{20 \text{ days}}{4 \text{ days/half-life}} = \textbf{5 half-lives}$ .

We know that after 20 days (5 half-lives), we have 4 mg left.
So, $\frac{1}{32}$ of the initial mass is equal to 4 mg.
Let's call the initial mass $m_0$ .
We can write this as an equation: $\frac{1}{32} \cdot m_0 = 4$ .

To find $m_0$ , we can multiply both sides of the equation by 32: $32 \cdot \frac{1}{32} \cdot m_0 = 4 \cdot 32$ .
This simplifies to $m_0 = \textbf{128 mg}$ .
So, the initial mass of the sample was 128 mg.

First, let's convert 4 weeks into days: $4 \text{ weeks} \cdot 7 \text{ days/week} = \textbf{28 days}$ .

Now, let's figure out how many half-lives are in 28 days: $\frac{28 \text{ days}}{4 \text{ days/half-life}} = \textbf{7 half-lives}$ .

After 7 half-lives, the remaining mass will be $(\frac{1}{2})^7 = \frac{1}{128}$ of the initial mass.

We found that the initial mass was 128 mg.
So, after 28 days (7 half-lives), the remaining mass will be $\frac{1}{128} \cdot 128 \text{ mg} = \textbf{1 mg}$ .

The initial mass of the Palladium-100 sample was 128 mg.
After 4 weeks, the remaining mass will be 1 mg.