# Use your equation to determine the half-life ofthis type of Fodine, That is, find out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs

Question
Modeling
Use your equation to determine the half-life ofthis type of Fodine, That is, find out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs

2020-12-01
To find the half life of iodine (i.e) $$t =\ ?\ when\ P=\frac{P_{0}}{2}=\ \frac{10}{2}=5$$ grams of iodine $$P = 10_{e}^{−0.086t}$$ substitute $$P = 5$$ in the above equation $$5 = 10_{e}^{−0.086t}$$ Dividing both sides by 10 we get, $$\frac{5}{10}=e^{-0.086t}$$
$$\Rightarrow\ e^{-0.086t}=\ \frac{1}{2}$$
$$e^{0.086t} = 2$$ Taking log on both sides we get, $$0.086t = \log_{e} 2$$
$$\Rightarrow\ t=\ \frac{\log_{e}2}{0.086}=\ \frac{0.69310}{0.086}=8.0598\ \approx\ 8$$ Therefore it took 8 days for the iodine reduces to 5 grams

### Relevant Questions

Determine the algebraic modeling a.
One type of Iodine disintegrates continuously at a constant rate of $$\displaystyle{8.6}\%$$ per day.
Suppose the original amount, $$\displaystyle{P}_{{0}}$$, is 10 grams, and let t be measured in days.
Because the Iodine is decaying continuously at a constant rate, we use the model $$\displaystyle{P}={P}_{{0}}{e}^{k}{t}$$ for the decay equation, where k is the rate of continuous decay.
Using the given information, write the decay equation for this type of Iodine.
b.
Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs.
Use your equation to determine the half-life ofthis type of lodine. That is, find out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs.
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a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
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c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
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Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is $$\displaystyle{\frac{{{d}{m}}}{{{\left.{d}{t}\right.}}}}=-{k}{m}$$ and the solution of that differential equation if $$\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{-{k}{t}}}$$, where $$\displaystyle{m}_{{0}}$$ is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
One type of Iodine disintegrates continuously at a constant rate of 8.6% per day. Suppose the original amount,$$P_0$$, is 10 grams, and let be measured in days. Because the Iodine is decaying continuously at a constant rate, we use the model $$P = P_0e^{kt}$$ for the decay equation, where k is the rate of continuous decay. Using the given information, write the decay equation for this type of Iodine.
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(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
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(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
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To model this, the city manager uses the function $$\displaystyle{C}{\left({t}\right)}={9.5}{\left({1}{042}\right)}^{t}\text{where}\ {C}{\left({t}\right)}$$ is the annual personnel costs,in millions of dollars, t years past 2009
1) Write the equation that can be solved to find in what year the personnel costs will be double the 2009 personnel costs.
2) Then solve the equation numerically (use the Table feature of your calculator) or graphically and determine the year.
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The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
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