The area, In square meters, of a pond covered by an algae bloom decreases exponentially after otreatment is applied. If the
area decreases from 150 to 75 over two days, how many days will it take for the algoe to cover after the treatment is
aplica?
0

Ask by Washington Barnett. in the United States

Mar 23,2025

Question

The area, In square meters, of a pond covered by an algae bloom decreases exponentially after otreatment is applied. If the
area decreases from 150 to 75 over two days, how many days will it take for the algoe to cover after the treatment is
aplica?
0

Ask by Washington Barnett. in the United States

Mar 23,2025

UpStudy AI · Accepted Answer

The area of the pond after treatment is modeled by the exponential decay function

$$
A(t)=A_0 e^{-kt}.
$$

**Step 1. Find the decay constant $$ k $$.**

We are given that initially, $$ A(0)=150 $$ square meters, and after 2 days, the area is $$ A(2)=75 $$ square meters. Thus,

$$
A(2)=150e^{-2k}=75.
$$

Solving for $$ k $$:

$$
e^{-2k}=\frac{75}{150}=\frac{1}{2}.
$$

Taking the natural logarithm on both sides:

$$
-2k=\ln\left(\frac{1}{2}\right)=-\ln 2,
$$

which gives

$$
k=\frac{\ln 2}{2}.
$$

**Step 2. Determine the time $$ t $$ when $$ A(t)=\frac{75}{\sqrt{2}} $$.**

We have

$$
150e^{-kt}=\frac{75}{\sqrt{2}}.
$$

Divide both sides of the equation by 150:

$$
e^{-kt}=\frac{75}{150\sqrt{2}}=\frac{1}{2\sqrt{2}}.
$$

Take the natural logarithm on both sides:

$$
-kt=\ln\left(\frac{1}{2\sqrt{2}}\right).
$$

Notice that

$$
\ln\left(\frac{1}{2\sqrt{2}}\right)=-\ln(2\sqrt{2}).
$$

Thus, we have:

$$
-kt=-\ln(2\sqrt{2}),
$$
$$
kt=\ln(2\sqrt{2}).
$$

Express $$\ln(2\sqrt{2})$$ in terms of $$\ln 2$$:

$$
\ln(2\sqrt{2})=\ln 2+\ln\sqrt{2}=\ln 2+\frac{1}{2}\ln2=\frac{3}{2}\ln2.
$$

So,

$$
kt=\frac{3}{2}\ln2.
$$

Substitute $$ k=\frac{\ln2}{2} $$ into the equation:

$$
\left(\frac{\ln2}{2}\right)t=\frac{3}{2}\ln2.
$$

Solve for $$ t $$:

$$
t=\frac{\frac{3}{2}\ln2}{\frac{\ln2}{2}} = \frac{3}{2}\ln2\cdot\frac{2}{\ln2}=3.
$$

Thus, it will take $$ \boxed{3} $$ days for the algae to cover $$\frac{75}{\sqrt{2}}$$ square meters after the treatment.
It will take 3 days for the algae to cover $$\frac{75}{\sqrt{2}}$$ square meters after the treatment.

The area, In square meters, of a pond covered by an algae bloom decreases exponentially after otreatment is applied. If the
area decreases from 150 to 75 over two days, how many days will it take for the algoe to cover after the treatment is
aplica?
0

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The area, In square meters, of a pond covered by an algae bloom decreases exponentially after otreatment is applied. If the area decreases from 150 to 75 over two days, how many days will it take for the algoe to cover after the treatment is aplica? 0

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Latest Pre Calculus Questions

The area, In square meters, of a pond covered by an algae bloom decreases exponentially after otreatment is applied. If the
area decreases from 150 to 75 over two days, how many days will it take for the algoe to cover after the treatment is
aplica?
0