High School

The number of people with the flu during an epidemic is a function [tex] f [/tex] of the number of days [tex] d [/tex] since the epidemic began. The equation [tex] f(d) = 50 \cdot \left(\frac{3}{2}\right)^d [/tex] defines [tex] f [/tex].

How many people had the flu at the beginning of the epidemic?

A. 150
B. [tex] \frac{3}{2} [/tex]
C. 50
D. 250

Answer :

To find out how many people had the flu at the beginning of the epidemic, we need to evaluate the function [tex]\( f(d) = 50 \cdot \left(\frac{3}{2}\right)^d \)[/tex] when the number of days [tex]\( d \)[/tex] is 0, because the beginning means no days have passed since the epidemic started.

Here's how you can do it step-by-step:

1. Understand the Function: The given function [tex]\( f(d) \)[/tex] represents the number of people with the flu as days go by. It’s in the form of an exponential growth function, where 50 is the initial number of people and [tex]\( \left(\frac{3}{2}\right)^d \)[/tex] represents the growth rate over time.

2. Set the Day to Start: Since we're looking for the number of people at the very start of the epidemic, we set [tex]\( d = 0 \)[/tex].

3. Substitute [tex]\( d = 0 \)[/tex] Into the Function: Replace [tex]\( d \)[/tex] with 0 in the equation:
[tex]\[
f(0) = 50 \cdot \left(\frac{3}{2}\right)^0
\][/tex]

4. Evaluate the Exponent: Any number raised to the power of 0 is 1. Therefore, [tex]\( \left(\frac{3}{2}\right)^0 = 1 \)[/tex].

5. Calculate the Function Value: Now the equation simplifies to:
[tex]\[
f(0) = 50 \cdot 1 = 50
\][/tex]

6. Conclusion: At the beginning of the epidemic, 50 people had the flu.

Thus, the correct answer is 50.