Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start,...

Question:

Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start, how many days until there are 1,000 mushrooms?

Exponential Growth

A quantity is said to grow exponentially if it increases by a common factor. A general equation for exponential growth is given by {eq}Q=ab^t {/eq} where {eq}a {/eq} is the initial value, {eq}Q {/eq} is the quantity at time {eq}t {/eq} and {eq}b {/eq} is the growth factor. Some examples of exponential growth are populations and compounding interest in investments.

Answer and Explanation:

This is an exponential growth problem, since the mushroom population triples every 10 days. We need to write an equation that will multiply our mushroom population by {eq}3 {/eq} every ten days. An exponential equation can be written in the form {eq}Q=ab^t {/eq}. In this problem, our initial value is {eq}a=10 {/eq} and if {eq}t=10 {/eq} we have {eq}Q=30 {/eq}. We can use this information to solve for the growth factor {eq}b {/eq}.


$$\begin{align*} Q&=ab^t\\ 30&=10b^{10}\\ 3&=b^{10}\\ \sqrt[10]{3}&=b \end{align*} $$


Now that we have the growth factor, we can write the general model for the mushroom colony growth as {eq}Q=10(\sqrt[10]{3})^t {/eq}. To find how many days until there are 1000 mushrooms, substitute {eq}Q=1000 {/eq}.


$$\begin{align*} Q&=10(\sqrt[10]{3})^t\\ 1000&=10(\sqrt[10]{3})^t\\ 100&=(\sqrt[10]{3})^t\\ \end{align*} $$

To solve for a variable that is in the exponent, we need to take the logarithm of both sides of the equation. The type of logarithm does not matter - we will use the common logarithm here.


$$\begin{align*} 100&=(\sqrt[10]{3})^t\\ \log(100)&=\log((\sqrt[10]{3})^t)\\ \log(100)&=t\log(\sqrt[10]{3})\\ \dfrac{\log(100)}{\log(\sqrt[10]{3})}&=t\\ t&\approx 41.92\\ \end{align*} $$

Therefore, It will take about 42 days for the mushroom colony to reach 1000 mushrooms.


Learn more about this topic:

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Exponential Growth: Definition & Examples

from High School Algebra I: Help and Review

Chapter 6 / Lesson 10
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