Question

a biologist is monitoring the hourly growth of bacteria when introduced to an anti - bacterial. the functions $f(t)=500(2^{t})$ and $g(t)=750(2^{t})$ model the growth rate of the same type of bacteria in two petri dishes for $t$ hours. the biologist adds the functions to determine the total number of bacteria during the first stage of the experiment. his sum is given below.
is the biologist’s sum correct? use the drop - down menus to explain.
$f(t)+g(t)=1250(4^{t})$

Explanation:

Step1: Identify common factor

Both terms have $2^t$ as a factor.

Step2: Factor out $2^t$

$f(t)+g(t) = 500(2^t) + 750(2^t) = (500+750)2^t$

Step3: Calculate the sum of coefficients

$500+750=1250$, so $f(t)+g(t)=1250(2^t)$

Step4: Compare to biologist's sum

The biologist claims $f(t)+g(t)=1250(4^t)$, but $4^t=(2^2)^t=2^{2t}
eq 2^t$.

Answer:

The biologist's sum is incorrect. The correct total growth rate function is $f(t)+g(t)=1250(2^t)$, not $1250(4^t)$, because we factor out the common exponential term $2^t$ rather than incorrectly combining the bases of the exponents.