# exponential functions

#### chassity

##### New member
The area of a wetland drops by a sixth every four years. What percent of its total area disappears after twenty years?

I cant figure this one out... i keep getting 99.987 which would be 99.99% and when i enter that it tells me its wrong. any help would be greatly appreciated.

#### Denis

##### Senior Member
HINT:
(1 - 1/6)^(20/4) = (5/6)^5 = .4018775....

#### soroban

##### Elite Member
Hello, chassity!

The area of a wetland drops by a sixth every four years.
What percent of its total area disappears after twenty years?

$$\displaystyle \text{Every 4 years, the wetland has only }\tfrac{5}{6}\text{ of its previous area.}$$

$$\displaystyle \text{The function is: }\:A \:=\:A_o\left(\tfrac{5}{6}\right)^\frac{x}{4}}\;\;\text{ where: }\:\begin{Bmatrix}A_o &=& \text{initial area} \\ x &=& \text{no. of years} \end{Bmatrix}$$

$$\displaystyle \text{For }x = 20\text{, we have: }\:\frac{A}{A_o} \:=\:\left(\tfrac{5}{6}\right)^5\:=\:0.401877572 \;\approx\;40.19\%$$

$$\displaystyle \text{After 20 years, only 40.19\% of the wetland remains.}$$

$$\displaystyle \text{Therefore, about 59.81\% of the total area has disappeared.}$$

#### lookagain

##### Senior Member
chassity said:
The area of a wetland drops by a sixth every four years.
What percent of its total area disappears after twenty years?

chassity,

for 20 years with a drop every 4 years, there are a total of 5 drops. After the first 4 years,

there is a drop by a $$\displaystyle \frac{1}{6}$$ of the amount, leaving $$\displaystyle \frac{5}{6}$$ of the amount remaining.

$$\displaystyle \frac{5}{6}A - \frac{1}{6}(\frac{5}{6}A)$$ leaves a drop by $$\displaystyle \frac{1}{6}(\frac{5}{6})$$ of the amount, and this is added to

the first drop by $$\displaystyle \frac{1}{6}$$ of the amount. Then there is $$\displaystyle \frac{25}{36}A$$ remaining after 8 years.

Continuing in this way, the sum of the drops for the total of 5 drops is

$$\displaystyle \frac{1}{6} + \frac{1}{6}( \frac{5}{6}) + . . . + \frac{1}{6}(\frac{5}{6})^4 \ = \$$

$$\displaystyle \frac{1}{6}[1 + \frac{5}{6} + . . . +(\frac{5}{6})^4] \ = \ ?$$

Please finish the calculation (including converting the result into a percent).