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Talja [164]
3 years ago
5

You have just isolated a new radioactive element. If you can determine its half-life, you will win the Nobel Prize in physics. Y

ou purify a sample of 2 grams. One of your colleagues steals half of it, and four days later you find that 0.3 gram of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.)
Mathematics
1 answer:
Troyanec [42]3 years ago
6 0
Staals half
1 gram left
0.3 grams left
half lives are normally written in years
it goes likt this

A=P( \frac{1}{2})^ \frac{t}{h}
A=final amount
P=initila amount
t=time elapsed
h=time half life is (same units as time)

let's use days as the units of time


we are given
initial amount=1gram
final amount=0.3gram
time=4 days

so
0.3=1( \frac{1}{2})^ \frac{4}{h}
solve for h
0.3=( \frac{1}{2})^ \frac{4}{h}
take the ln of both sides
ln(0.3)=ln( \frac{1}{2})^ \frac{4}{h}
ln(0.3)=\frac{4}{h}ln( \frac{1}{2})
times both sides by h
hln(0.3)=4ln( \frac{1}{2})
divide both sides by ln(0.3)
h= \frac{4ln( \frac{1}{2}) }{ln(0.3)}
use calculator
h=2.30287
3 sig figs
h=2.303 days
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A researcher wishes to estimate the average blood alcohol concentration​ (BAC) for drivers involved in fatal accidents who are f
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Answer:

A 90% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC is [0.143, 0.177] .

Step-by-step explanation:

We are given that a researcher randomly selects records from 60 such drivers in 2009 and determines the sample mean BAC to be 0.16 g/dL with a standard deviation of 0.080 ​g/dL.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

                               P.Q.  =  \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }  ~   t_n_-_1

where, \bar X = sample mean BAC = 0.16 g/dL

            s = sample standard deviation = 0.080 ​g/dL

            n = sample of drivers = 60

            \mu = population mean BAC in fatal crashes

<em>Here for constructing a 90% confidence interval we have used a One-sample t-test statistics because we don't know about population standard deviation. </em>

So, a 90% confidence interval for the population mean, \mu is;

P(-1.672 < t_5_9 < 1.672) = 0.90  {As the critical value of t at 59 degrees of

                                              freedom are -1.672 & 1.672 with P = 5%}    P(-1.672 < \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } } < 1.672) = 0.90

P( -1.672 \times {\frac{s}{\sqrt{n} } } < {\bar X-\mu} < 1.672 \times {\frac{s}{\sqrt{n} } } ) = 0.90

P( \bar X-1.672 \times {\frac{s}{\sqrt{n} } } < \mu < \bar X+1.672 \times {\frac{s}{\sqrt{n} } } ) = 0.90

<u>90% confidence interval for</u> \mu = [ \bar X-1.672 \times {\frac{s}{\sqrt{n} } } , \bar X+1.672 \times {\frac{s}{\sqrt{n} } } ]

                                       = [ 0.16-1.672 \times {\frac{0.08}{\sqrt{60} } } , 0.16+1.672 \times {\frac{0.08}{\sqrt{60} } } ]

                                       = [0.143, 0.177]

Therefore, a 90% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC is [0.143, 0.177] .

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After years of overhunting, environmental scientists have reintroduced mountain goats into Yellowstone National Park. The initia
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We want to see how much the population of goats grows each year. We will see that the correct option is c: 17%.

<h3>Exponential growth of populations</h3>

We know that:

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The population can be modeled with an exponential equation as:

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So we have:

P(t) = 1500*(1 + r)^t

And we know that after 11 years the population is 8,400, so we have:

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Now we can solve this for r:

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r = 0.17

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This means that the population increases a 17% each year, so the correct option is c.

If you want to learn more about exponential growth, you can read:

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