Math / Calculus

Question27. Find the half-life of a certain element that vaporizes in the air. If it has an initial volume of 50 mL and vaporized 12 mL in 12 minutes. View answer choices
28. A bacteria was placed inside a 500 mL bottle. Initially, the bacteria has filled up ii. 10 mL of the bottle. After 24 hours, the bottle is completely infested by the bacteria. Find the doubling time of the bacteria: $=$ t View answer choices
29. A radioactive mass was found at 8 am and weighed 50 grams. At 9:30 am it weighed 20 grams. Find the half-life of the mass in hrs. View answer choices
30. An ice block was delivered at 5 am and weighed 110 kg . At 10 am , the block was reduced to 70 kg . Find its half-life. View answer choices

Studdy Solution

STEP 1

What is this asking? How long does it take for half of the element's volume to disappear? Watch out! Don't mix up the remaining volume with the vaporized volume!

STEP 2

1. Set up the exponential decay formula
2. Plug in the given values
3. Solve for the half-life

STEP 3

We're dealing with exponential decay, so let's use the formula $V(t) = V_0 \cdot (1 - r)^t$ , where $V(t)$ is the volume at time $t$ , $V_0$ is the initial volume, $r$ is the rate of decay, and $t$ is the time.
Remember, this formula tells us how much volume is left, not how much has vaporized!

STEP 4

We're given the time in minutes, so we'll express the rate in "amount vaporized per minute."

STEP 5

We know the initial volume $V_0 = 50$ mL.

STEP 6

After 12 minutes, 12 mL vaporized, so the remaining volume is $50 - 12 = 38$ mL.
So, $V(12) = 38$ .

STEP 7

The element vaporized 12 mL in 12 minutes, so the rate of decay per minute is $\frac{12}{50 \cdot 12} = \frac{1}{50}$ .
Notice how we divide by the initial volume to get the fraction of the volume that vaporized per minute.

STEP 8

Now, plug everything into our formula: $38 = 50 \cdot (1 - \frac{1}{50})^{12}$ .

STEP 9

We want to find the half-life, which is the time it takes for half the initial volume to vaporize.
So we set $V(t)$ to half of the initial volume: $\frac{50}{2} = 50 \cdot (1 - \frac{1}{50})^t$ .

STEP 10

Divide both sides by 50: $\frac{1}{2} = (1 - \frac{1}{50})^t$ or $\frac{1}{2} = (\frac{49}{50})^t$ .

STEP 11

Take the logarithm of both sides (you can use any base, but let's use base 10): $\log(\frac{1}{2}) = t \cdot \log(\frac{49}{50})$ .

STEP 12

Now, solve for $t$ : $t = \frac{\log(\frac{1}{2})}{\log(\frac{49}{50})}$ .

STEP 13

Calculate the value of $t$ : $t \approx \frac{-0.301}{-0.00877} \approx 34.32$ .
So, the half-life is approximately 34.32 minutes.

STEP 14

The half-life of the element is approximately 34.32 minutes.

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Studdy Solution

STEP 1

What is this asking? How long does it take for half of the element's volume to disappear? Watch out! Don't mix up the remaining volume with the vaporized volume!

STEP 2

1. Set up the exponential decay formula
2. Plug in the given values
3. Solve for the half-life

STEP 3

We're dealing with exponential decay, so let's use the formula $V(t) = V_0 \cdot (1 - r)^t$ , where $V(t)$ is the volume at time $t$ , $V_0$ is the initial volume, $r$ is the rate of decay, and $t$ is the time.
Remember, this formula tells us how much volume is left, not how much has vaporized!

STEP 4

We're given the time in minutes, so we'll express the rate in "amount vaporized per minute."

STEP 5

We know the initial volume $V_0 = 50$ mL.

STEP 6

After 12 minutes, 12 mL vaporized, so the remaining volume is $50 - 12 = 38$ mL.
So, $V(12) = 38$ .

STEP 7

The element vaporized 12 mL in 12 minutes, so the rate of decay per minute is $\frac{12}{50 \cdot 12} = \frac{1}{50}$ .
Notice how we divide by the initial volume to get the fraction of the volume that vaporized per minute.

STEP 8

Now, plug everything into our formula: $38 = 50 \cdot (1 - \frac{1}{50})^{12}$ .

STEP 9

We want to find the half-life, which is the time it takes for half the initial volume to vaporize.
So we set $V(t)$ to half of the initial volume: $\frac{50}{2} = 50 \cdot (1 - \frac{1}{50})^t$ .

STEP 10

Divide both sides by 50: $\frac{1}{2} = (1 - \frac{1}{50})^t$ or $\frac{1}{2} = (\frac{49}{50})^t$ .

STEP 11

Take the logarithm of both sides (you can use any base, but let's use base 10): $\log(\frac{1}{2}) = t \cdot \log(\frac{49}{50})$ .

STEP 12

Now, solve for $t$ : $t = \frac{\log(\frac{1}{2})}{\log(\frac{49}{50})}$ .

STEP 13

Calculate the value of $t$ : $t \approx \frac{-0.301}{-0.00877} \approx 34.32$ .
So, the half-life is approximately 34.32 minutes.

STEP 14

The half-life of the element is approximately 34.32 minutes.

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Studdy Solution

STEP 1

What is this asking? How long does it take for half of the element's volume to disappear? Watch out! Don't mix up the *remaining* volume with the *vaporized* volume!

STEP 2

1. Set up the exponential decay formula2. Plug in the given values3. Solve for the half-life

STEP 3

STEP 4

We're given the time in minutes, so we'll express the rate in "amount vaporized per minute."

STEP 5

We know the **initial volume** V0=50V_0 = 50V0​=50 mL.

STEP 6

After 12 minutes, 12 mL vaporized, so the remaining volume is 50−12=3850 - 12 = 3850−12=38 mL.So, V(12)=38V(12) = 38V(12)=38.

STEP 7

The element vaporized 12 mL in 12 minutes, so the rate of decay per minute is 1250⋅12=150\frac{12}{50 \cdot 12} = \frac{1}{50}50⋅1212​=501​.Notice how we divide by the **initial volume** to get the *fraction* of the volume that vaporized per minute.

STEP 8

Now, plug everything into our formula: 38=50⋅(1−150)1238 = 50 \cdot (1 - \frac{1}{50})^{12}38=50⋅(1−501​)12.

STEP 9

We want to find the half-life, which is the time it takes for half the initial volume to vaporize.So we set V(t)V(t)V(t) to half of the initial volume: 502=50⋅(1−150)t\frac{50}{2} = 50 \cdot (1 - \frac{1}{50})^t250​=50⋅(1−501​)t.

STEP 10

Divide both sides by 50: 12=(1−150)t\frac{1}{2} = (1 - \frac{1}{50})^t21​=(1−501​)t or 12=(4950)t\frac{1}{2} = (\frac{49}{50})^t21​=(5049​)t.

STEP 11

Take the logarithm of both sides (you can use any base, but let's use base 10): log⁡(12)=t⋅log⁡(4950)\log(\frac{1}{2}) = t \cdot \log(\frac{49}{50})log(21​)=t⋅log(5049​).

STEP 12

Now, solve for ttt: t=log⁡(12)log⁡(4950)t = \frac{\log(\frac{1}{2})}{\log(\frac{49}{50})}t=log(5049​)log(21​)​.

STEP 13

Calculate the value of ttt: t≈−0.301−0.00877≈34.32t \approx \frac{-0.301}{-0.00877} \approx 34.32t≈−0.00877−0.301​≈34.32.So, the half-life is approximately **34.32 minutes**.

STEP 14

The half-life of the element is approximately **34.32 minutes**.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? How long does it take for half of the element's volume to disappear? Watch out! Don't mix up the remaining volume with the vaporized volume!

1. Set up the exponential decay formula
2. Plug in the given values
3. Solve for the half-life

We know the initial volume $V_0 = 50$ mL.

After 12 minutes, 12 mL vaporized, so the remaining volume is $50 - 12 = 38$ mL.
So, $V(12) = 38$ .

The element vaporized 12 mL in 12 minutes, so the rate of decay per minute is $\frac{12}{50 \cdot 12} = \frac{1}{50}$ .
Notice how we divide by the initial volume to get the fraction of the volume that vaporized per minute.

Now, plug everything into our formula: $38 = 50 \cdot (1 - \frac{1}{50})^{12}$ .

We want to find the half-life, which is the time it takes for half the initial volume to vaporize.
So we set $V(t)$ to half of the initial volume: $\frac{50}{2} = 50 \cdot (1 - \frac{1}{50})^t$ .

Divide both sides by 50: $\frac{1}{2} = (1 - \frac{1}{50})^t$ or $\frac{1}{2} = (\frac{49}{50})^t$ .

Take the logarithm of both sides (you can use any base, but let's use base 10): $\log(\frac{1}{2}) = t \cdot \log(\frac{49}{50})$ .

Now, solve for $t$ : $t = \frac{\log(\frac{1}{2})}{\log(\frac{49}{50})}$ .

Calculate the value of $t$ : $t \approx \frac{-0.301}{-0.00877} \approx 34.32$ .
So, the half-life is approximately 34.32 minutes.

The half-life of the element is approximately 34.32 minutes.