Question 1101289
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The number of bacteria is increasing exponentially, so the function is of the form
{{{y = ab^x}}}
where a is an initial value (when x=0) and b is the growth rate.<br>
(1) Use the two given pieces of information to write equations in that form, using x as the time in hours:
{{{200 = ab^2}}}
{{{25000 = ab^5}}}<br>
(2) Divide the second equation by the first; that will eliminate a, allowing you to find the value of b:
{{{125 = b^3}}}
{{{5 = b}}}<br>
Remember not to just go through the motions of solving the problem; take the time to realize that b=5 means the number of bacteria grows by a factor of 5 each hour.
(You will appreciate the mathematics more if you understand what it is doing....)<br>
(3) Use the value you have found for b in either of the original equations to find the value of a:
{{{200 = a(5)^2}}}
{{{200 = 25a}}}
{{{a = 8}}}<br>
So the function is {{{y = 8(5)^x}}}<br>
For part b, you need to solve the equation {{{8(5)^x = 2000}}}<br>
Up to now, the numbers have worked out nicely; a and b are whole numbers.  That will no longer be the case:
f(0) = 8
f(1) = 40
f(2) = 200
f(3) = 1000
f(4) = 5000
f(5) = 25000
So the solution to part b will be between 3 and 4 hours.<br>
To solve algebraically, because the variable is in an exponent, you need to use logarithms.<br>
{{{8(5^x) = 2000}}}
{{{5^x = 250}}}
{{{x*log(5) = log(250)}}}
{{{x = log(250)/log(5) = 3.43}}}  (to 2 decimal places)