Question

a biologist is monitoring the hourly growth of bacteria when introduced to an anti-bacterial. the functions $f(t) = 500\left(2^t\
ight)$ and $g(t) = 750\left(2^t\
ight)$ 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.

$f(t) + g(t) = 1250\left(4^t\
ight)$

click the arrows to choose an answer from each menu.

to add the functions, the biologist should choose... add the coefficients of the exponential expressions. the sum of the functions is $f(t) + g(t) =$ choose...
the biologist’s sum choose... correct.

Explanation:

Step1: Analyze the functions

We have \( f(t) = 500(2^t) \) and \( g(t)=750(2^t) \). When adding two exponential functions with the same base and exponent, we can factor out the common exponential term.

Step2: Add the functions

Using the distributive property \( a\times c + b\times c=(a + b)\times c \), here \( c = 2^t \), \( a = 500 \), \( b=750 \). So \( f(t)+g(t)=(500 + 750)2^t=1250(2^t) \). The biologist incorrectly converted \( 2^t+2^t \) to \( 4^t \), but we should only add the coefficients when the bases and exponents of the exponential terms are the same.

Step3: Evaluate the biologist's sum

The biologist's sum is \( 1250(4^t) \), but the correct sum is \( 1250(2^t) \), so the biologist's sum is not correct.

Answer:

To add the functions, the biologist should \(\boldsymbol{\text{add the coefficients}}\) of the exponential expressions. The sum of the functions is \( f(t)+g(t)=\boldsymbol{1250(2^t)} \). The biologist's sum \(\boldsymbol{\text{is not}}\) correct.