Question

I need help with this question

Answer 1

The number of cows is given by

$14\cdot 2^{\frac{y}{5}}$

So, after $k$ years, the number of cows will be

$14\cdot 2^{\frac{y+k}{5}}$

We want this number to be twice as much as the original:

$14\cdot 2^{\frac{y+k}{5}} = 2(14\cdot 2^{\frac{y}{5}})$

First of all, we can cancel 14 from both sides:

$2^{\frac{y+k}{5}} = 2\cdot 2^{\frac{y}{5}}$

Finally, on the right hand side, we can use the exponent rule

$a^b\cdot a^c=a^{b+c}$

to get

$2^{\frac{y+k}{5}} = 2^{\frac{y}{5}+1}$

To solve this equation, we must impose that the two exponents are the same:

$\dfrac{y+k}{5} = \dfrac{y}{5}+1 \iff \dfrac{y+k}{5} = \dfrac{y+5}{5}$

And clearly this is true if and only if $k=5$. So, it will take 5 years for the cow heard to double in number.

You can do the exact same steps to find the doubling time for the sheeps.