Question

The landscaper pours 200 gallons of herbicide in a pond. The herbicide degrades 11% each week.Write an equation to find the amount of herbicide in any given week.How much will be in the pond after 2 weeks?The landscaper will put another dose in the pond when the herbicide level drops below 50 gallons. In about how many weeks will he need to add more herbicide?

Answer 1

To answer this question, we will use the following general form of an exponential model:

`T=P(1\pm r)^t,`

where P is the initial amount, r is the rate in decimal form, and t is the time.

In this case, since the herbicide degrades, the sign inside the parenthesis will be a minus sign, P=200, r=0.11, and t will be the time in weeks, therefore the number of gallons after t weeks can be modeled by the following equation:

`T=200(1-0.11)^t.`

Evaluating the above equation at t=2, we get:

(Answer part 1)

`T=200(1-0.11)^2=158.42.`

If we set T=50, and solve for t, we get:

(Answer part 2)

`\begin{gathered} 50=200(1-0.11)^t, \\ (50)/(200)=(0.89)^t, \\ (1)/(4)=0.89^t, \\ \ln ((1)/(4))=t\ln 0.89, \\ t=(\ln ((1)/(4)))/(\ln 0.89)\approx12. \end{gathered}`

Answer:

There will be 158 gallons after 2 weeks.

In about 12 weeks the landscaper has to put another dose.