# Find the number of units of the molecule in the river after n weeks

• Imuell1
In summary, the number of units of a molecule in a river after n weeks can be determined using a mathematical formula that considers factors such as initial amount, introduction and removal rates, and environmental conditions. The amount can be affected by various factors and can decrease over time if removal rates are higher. While predictions can be made, they may not always be accurate due to uncertainties. Knowing the amount of the molecule can be useful in understanding its impact on the environment and human health, regulating its introduction, and tracking the effectiveness of pollution control measures.
Imuell1

## Homework Statement

A chemical plant produces pesticide that contains a molecule potentially harmful to people if the concentration is too high. The plant flushes out the tanks containing the pesticide once a week, and the discharge flows into the river. The molecule breaks down gradually in water so that 90% of the amount remaining each week is dissipated by the end of the next week. Suppose that D units of the molecule are discharged each week.

a) Find the number of units of the molecule in the river after n weeks.
b) Estimate the amount of the molecule in the water supply over a very long time(hint find the sum of the series)
c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?

## The Attempt at a Solution

A)
I got a1=D a2=D+.01(D) a3=D+.01(D+.01D)

so an=D+.01(an-1) or an=a1+.01(an-1)

B) I'm not sure if I did part b right and I got the sum as n=0 to $$\infty$$ of a1+.01(an-1) but I have a strong feeling this isn't right.

C) I don't know what to do for part C

Imuell1 said:

## Homework Statement

A chemical plant produces pesticide that contains a molecule potentially harmful to people if the concentration is too high. The plant flushes out the tanks containing the pesticide once a week, and the discharge flows into the river. The molecule breaks down gradually in water so that 90% of the amount remaining each week is dissipated by the end of the next week. Suppose that D units of the molecule are discharged each week.

a) Find the number of units of the molecule in the river after n weeks.
b) Estimate the amount of the molecule in the water supply over a very long time(hint find the sum of the series)
c) If the toxic level of the molecule is T units, how large an amount of the molecule can the plant discharge each week?

## The Attempt at a Solution

A)
I got a1=D a2=D+.01(D) a3=D+.01(D+.01D)

so an=D+.01(an-1) or an=a1+.01(an-1)
This is a "recursive" equation but it does not answer the question- you have not yet found a formula for an. You say that a3= D(1+ .01+ .012). Do you see that that is a geometric series? What is the formula for the sum of a finite geometric series?

B) I'm not sure if I did part b right and I got the sum as n=0 to $$\infty$$ of a1+.01(an-1) but I have a strong feeling this isn't right.
Well, what is that sum? What is the formula for the sum of an infinite geometric series?

C) I don't know what to do for part C
For what values of D is the answer to (B) less than T?

## 1. How do you determine the number of units of the molecule in the river after n weeks?

To determine the number of units of the molecule in the river after n weeks, we use a mathematical formula that takes into account the initial amount of the molecule, the rate at which it is being introduced into the river, and the rate at which it is being removed from the river. This formula is based on the principles of exponential growth and decay.

## 2. What factors can affect the number of units of the molecule in the river after n weeks?

The number of units of the molecule in the river after n weeks can be affected by various factors such as the initial amount of the molecule, the rate at which it is being introduced into the river (e.g. through industrial discharge), and the rate at which it is being removed from the river (e.g. through natural processes or water treatment). Other factors such as weather patterns and environmental conditions can also play a role.

## 3. Is it possible for the number of units of the molecule in the river to decrease over time?

Yes, it is possible for the number of units of the molecule in the river to decrease over time. This can occur if the rate at which it is being removed from the river is higher than the rate at which it is being introduced. Additionally, natural processes such as dilution or degradation can also contribute to a decrease in the number of units of the molecule.

## 4. Can the number of units of the molecule in the river after n weeks be accurately predicted?

While we can use mathematical models to estimate the number of units of the molecule in the river after n weeks, it is important to note that there are many variables and uncertainties involved. Therefore, the prediction may not always be completely accurate. It is important to continually monitor and adjust the model as new data becomes available.

## 5. How can knowing the number of units of the molecule in the river after n weeks be useful?

Knowing the number of units of the molecule in the river after n weeks can be useful in determining the impact of the molecule on the environment and potential risks to human health. It can also help inform decisions on regulating the amount of the molecule being introduced into the river and implementing mitigation measures to reduce its presence in the water. Additionally, it can aid in tracking the effectiveness of pollution control measures over time.

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