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 biologists sum correct? use the drop - down menus to explain. $f(t)+g(t)=1250(4^{t})$ 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 biologists sum is not options: only; add the bases of the functions and; add the exponents of the functions and

Explanation:

Step1: Analyze the functions

We have \( f(t) = 500(2^t) \) and \( g(t) = 750(2^t) \). These are exponential functions with the same base (\( 2^t \)) and different coefficients (500 and 750).

Step2: Add the functions

When adding two exponential functions with the same base and exponent, we use the distributive property (similar to combining like terms). So, \( f(t)+g(t)=500(2^t)+750(2^t)=(500 + 750)(2^t) \).

Step3: Calculate the sum of coefficients

Calculate \( 500+750 = 1250 \). So, \( f(t)+g(t)=1250(2^t) \), not \( 1250(4^t) \). The biologist incorrectly changed the base from \( 2^t \) to \( 4^t \) (which would be equivalent to multiplying the exponents or bases incorrectly) instead of just adding the coefficients of the like - exponential terms.

Answer:

To add the functions, the biologist should \(\boldsymbol{\text{only}}\) add the coefficients of the exponential expressions \( 500(2^t) \) and \( 750(2^t) \). The biologist's sum \(\boldsymbol{\text{is not}}\) correct because when adding \( 500(2^t)+750(2^t) \), we get \( (500 + 750)(2^t)=1250(2^t) \), not \( 1250(4^t) \).