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3 years ago

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A radioactive substance decays exponentially: The mass at time t is m(t) = m(0)e^kt, where m(0) is the initial mass and k is a n

egative constant. The mean life M of an atom in the substance is
[infinity]
M = âk â« te^kt dt.
0

For the radioactive carbon isotope, 14C, used in radiocarbon dating, the value of k is -0.000121. Find the mean life of a 14C atom.

1 answer:

forsale [732]3 years ago

5 0

Answer:

mean life = 8264.5 s

Step-by-step explanation:

k = - 0.000121

The relation is given by

$m = mo e^{kt}$

Now, the mean life is the life time for which the sample retains.

The mean life is the reciprocal of the decay constant.

The relation between the mean life and the decay constant is

$\tau =\frac{1}{k}\\\\\tau = \frac{1}{0.000121} = 8264.5 seconds$

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Problem 5.A miner is trapped inside a mine and has access to 3 doors. The first door leads to a tunneland can take him to safety

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Answer:

Expected time is 15 hours for him to get to safety.

Step-by-step explanation:

We define X as the time that this miner would get to safety.

We define Y as the door he chooses initially.

P(Y= 1) = P(Y=2)=P(Y=3) = 1/3

We have E[X|Y=1] = 3

E[X|Y] = 5 hours + E[X}

E[X|Y] = 7 hours + E[X]

Then we have

E[X] = 1/3(3 + 5 + E[X] + 7 + E[X])

We cross multiply

3*E[X] = (15 + 2E[x])

3E[X] - 2E[X] = 15

E[X] = 15

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3 years ago

In October of 2012, Apple introduced a much smaller variant of the Apple iPad, known at the iPad Mini. Weighing less than 11 oun

sveticcg [70]

Answer:

a. $f_X(x) = \dfrac{1}{3.5}8.5$

b. the probability that the battery life for an iPad Mini will be 10 hours or less is 0.4286 which is about 42.86%

c. the probability that the battery life for an iPad Mini will be at least 11 hours is 0.2857 which is about 28.57 %

d. the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours is 0.5714 which is about 57.14%

e. 86 should have a battery life of at least 9 hours

Step-by-step explanation:

From the given information;

Let X represent the continuous random variable with uniform distribution U (A, B) . Therefore the probability density function can now be determined as :

$f_X(x) = \dfrac{1}{B-A}A$

where A and B are the two parameters of the uniform distribution

From the question;

Assume that battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours

So; Let A = 8,5 and B = 12

Therefore; the mathematical expression for the probability density function of battery life is :

$f_X(x) = \dfrac{1}{12-8.5}8.5$

$f_X(x) = \dfrac{1}{3.5}8.5$

b. What is the probability that the battery life for an iPad Mini will be 10 hours or less (to 4 decimals)?

The probability that the battery life for an iPad Mini will be 10 hours or less can be calculated as:

F(x) = P(X ≤x)

$F(x) = \dfrac{x-A}{B-A}$

$F(10) = \dfrac{10-8.5}{12-8.5}$

F(10) = 0.4286

the probability that the battery life for an iPad Mini will be 10 hours or less is 0.4286 which is about 42.86%

c. What is the probability that the battery life for an iPad Mini will be at least 11 hours (to 4 decimals)?

The battery life for an iPad Mini will be at least 11 hours is calculated as follows:

$P(X\geq11) = \int\limits^{12}_{11} {\dfrac{1}{3.5}} \, dx$

$P(X\geq11) = {\dfrac{1}{3.5}} (x)^{12}_{11}$

$P(X\geq11) = {\dfrac{1}{3.5}} (12-11)$

$P(X\geq11) = {\dfrac{1}{3.5}} (1)$

$P(X\geq11) = 0.2857$

the probability that the battery life for an iPad Mini will be at least 11 hours is 0.2857 which is about 28.57 %

d. What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours (to 4 decimals)?

$P(9.5 \leq X\leq11.5) =\int\limits^{11.5}_{9.5} {\dfrac{1}{3.5}} \, dx$

$P(9.5 \leq X\leq11.5) ={\dfrac{1}{3.5}} \, (x)^{11.5}_{9.5}$

$P(9.5 \leq X\leq11.5) ={\dfrac{1}{3.5}} (11.5-9.5)$

$P(9.5 \leq X\leq11.5) ={\dfrac{1}{3.5}} (2)$

$P(9.5 \leq X\leq11.5) =0.2857* (2)$

$P(9.5 \leq X\leq11.5) =0.5714$

Hence; the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours is 0.5714 which is about 57.14%

e. In a shipment of 100 iPad Minis, how many should have a battery life of at least 9 hours (to nearest whole value)?

The probability that battery life of at least 9 hours is calculated as:

$P(X \geq 9) = \int\limits^{12}_{9} {\dfrac{1}{3.5}} \, dx$

$P(X \geq 9) = {\dfrac{1}{3.5}}(x)^{12}_{9}$

$P(X \geq 9) = {\dfrac{1}{3.5}}(12-9)$

$P(X \geq 9) = {\dfrac{1}{3.5}}(3)$

$P(X \geq 9) = 0.2857*}(3)$

$P(X \geq 9) = 0.8571$

NOW; The Number of iPad that should have a battery life of at least 9 hours is calculated as:

n = 100(0.8571)

n = 85.71

n ≅ 86

Thus , 86 should have a battery life of at least 9 hours

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3 years ago

Karen purchased a prepaid phone card for $30. Long distance calls cost 16 cents a minute using this card. Karen used her card on

Sedbober [7]

30-24.08=5.92, 5.92/.16=37 minutes

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3 years ago

The sum of Joy's age and her brother's age is 24. Joy's brother's age is 3 years older than twice joy's age. What is the differe

enyata [817]

Let Joy be x years old

<span>Joy's brother's age is 3 years older than twice joy's age.
</span>Joy's Brother = 2x + 3

<span>The sum of Joy's age and her brother's age is 24.
x + 2x + 3 = 24
3x + 3 = 24
3x = 24 - 3
3x = 21
x = 21 </span>÷ 3
x = 7

Joy:
x = 7

Brother:
2x + 3 = 2(7) + 3 = 17

Difference in their age = 17 - 7 = 10 years

They are 10 years apart.

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4 years ago