Question 712352
Let {{{ S }}} = population of Sprogs
Let {{{ G }}} = population of Groks
Let {{{n }}} = number of years after initial count
For Sprogs, the formula is:
{{{ S = 1600*2^( n/14 ) }}}
For Graks,
{{{ G = 200*2^( n/7 ) }}}
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(a)
After {{{ 3.5 }}} years,
{{{ G = 200*2^(3.5/7) }}}
{{{ G = 200*2^.5 }}}
{{{ G = 200*1.414 }}}
{{{ G = 282.8 }}} ( you can add the extra decimals )
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When is {{{ S = G }}} ?
{{{ 1600*2^( n/14 ) = 200*2^( n/7 ) }}}
{{{ 8*2^( n/14 ) = 2^( n/7 ) }}}
{{{ 2^3*2^( n/14 ) = 2^( n/7 ) }}}
{{{ 2^( n/14 + 3 ) = 2^( n/7 ) }}}
{{{ n/14 + 3 = n/7 }}}
Multiply both sides by {{{ 14 }}}
{{{ n + 42 = 2n }}}
{{{ n = 42 }}}
In 42 years, their populations will be the same
check:
{{{ 1600*2^( 42/14 ) = 200*2^( 42/7 ) }}}
{{{ 1600*2^3 = 200*2^6 }}}
{{{ 1600*8 = 200*64 }}}
{{{ 12800 = 12800 }}}
OK