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

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] at the start, which means we look at when [tex]\( d = 0 \)[/tex].

Here's how we determine the number of people at the beginning:

1. Identify the Starting Point: The epidemic begins at day [tex]\( d = 0 \)[/tex].

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

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

4. Multiply by the Initial Value: Multiply 50 by 1:
[tex]\[
f(0) = 50 \cdot 1 = 50
\][/tex]

Therefore, at the beginning of the epidemic, 50 people had the flu. Hence, the answer is 50.