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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.
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.
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