# A riverbank is eroding exponentially so that every year it loses 5% of its soil. How much soil...

## Question:

A riverbank is eroding exponentially so that every year it loses 5% of its soil. How much soil will it have in 10 years?

## Exponential Growth:

In mathematics, exponential growth is a way that a quantity may increase over time. If growth is exponential, then the instantaneous rate of change of the quantity over time is proportional to the quantity itself.

Given data:

• The rate is {eq}r = 5\% = 0.05 {/eq}
• The time is {eq}t = 10\,{\rm{years}} {/eq}

The expression for exponential decay is given by

{eq}A = {A_0}{\left( {1 - r} \right)^t} {/eq}

• {eq}A {/eq} is the amount after t years
• {eq}{A_0} {/eq} is the initial amount

Substituting the values in the above equation:

{eq}\begin{align*} A &= {A_0}{\left( {1 - r} \right)^t}\\ A &= {A_0}{\left( {1 - 0.05} \right)^{10}}\\ A &= {A_0}{\left( {0.95} \right)^{10}}\\ A &= 0.5987\,{A_0}\\ A &= 59.87\% {A_0} \end{align*} {/eq}

Thus the soil will it have in 10 years is {eq}A = 59.87\% {A_0} {/eq}.