Question

If a population doubles every 5 years, how many years will it take for the population to increase by 10 times it's original amount?

Answer 1

Answer:

16.61 years.

Step-by-step explanation:

Let P be the original population and t be the time in years. We have been given that a population doubles every 5 years, which means that our population is increasing exponentially.

We can write an equation for our exponential function as:

$P(t)=P2^{\frac{t}{5}}$  

To find the time after which our population will be 10 times its original amount let us substitute P(t)=10P is our equation.

$10P = P(2)^{\frac{t}{5}}$

$10 = 2^{\frac{t}{5}}$

Now let us solve for t.

Upon taking natural log of both sides we will get,

$ln 10 =ln 2^{\frac{t}{5}}$

$ln 10 =\frac{t}{5}\cdot ln 2$

$2.3025850929940457 =\frac{t}{5}\cdot 0.6931471805599453$

$t=5\times \frac{2.3025850929940457}{0.6931471805599453}$

$t=5\times 3.3219280948873624161$  

$t=16.6096404744368120805\approx 16.61$

Therefore, it will take 16.61 years for the population to increase by 10 times it's original amount.

Answer 2

Answer:

After ≈$16.61$ years the population will increased by $10$ times it's original amount.

Step-by-step explanation:

Given: If a population doubles every $5$ years.

Let $A$ be the original population and $t$ be the time in years.

As per question given that a population doubles every $5$ years, which means that our population is increasing exponentially.

Here, the equation can be written as follows: $A(t)=A(2)^\frac{t}{5}$      .........(1)

Now, we have to find the time after which the population will be $10$ times its original amount.

Let us substitute $A(t)=10A$ in the above equation.

$10A=A(2)^\frac{t}{5}$

  $10=(2)^\frac{t}{5}$

Taking the natural log of both sides we get:

$ln10=\frac{t}{5}\times{ln2}\; \;\; \; \;\{ {ln(a)^n=nln(a)}}\} \\2.3025\times5=t\times 0.6931\\\\t=\frac{2.3025}{0.6931}\\t=16.6096$

Therefore, After ≈$16.61$ years the population will increased by $10$ times it's original amount.

Learn more about Exponential function here:

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