Question

a marketing specialist determines that when a certain product is released, the number of social media references to it can be modeled by
$f(t)=900 + 5\log _{5}(t)$
where $t$ is the number of days since the release.
determine $f(t)=$
$f(3)=$
which of the following is the best interpretation of $f(3)=903$
select an answer
which of the following is the best interpretation of $f(3)$?
select an answer
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Explanation:

Step1: Differentiate \( f(t) \)

Use the formula \( \frac{d}{dt}\log_a(t)=\frac{1}{t\ln(a)} \).
For \( f(t) = 900+5\log_5(t) \), the derivative of a constant \( 900 \) is \( 0 \).
The derivative of \( 5\log_5(t) \) is \( 5\times\frac{1}{t\ln(5)}=\frac{5}{t\ln(5)} \). So \( f'(t)=\frac{5}{t\ln(5)} \).

Step2: Calculate \( f'(3) \)

Substitute \( t = 3 \) into \( f'(t) \).
\( f'(3)=\frac{5}{3\ln(5)} \). Using a calculator, \( \ln(5)\approx1.6094 \), then \( f'(3)=\frac{5}{3\times1.6094}\approx1.039 \).

Answer:

\( f'(t)=\frac{5}{t\ln(5)} \)
\( f'(3)\approx1.039 \)

For the interpretation of \( f(3) = 903 \): Since \( f(t) \) models the number of social - media references to the product and \( t \) is the number of days since release, \( f(3)=903 \) means that 3 days after the product's release, there are 903 social - media references to the product.

For the interpretation of \( f'(3) \): Since \( f'(t) \) is the derivative of \( f(t) \) (the rate of change of the number of social - media references with respect to time \( t \)), \( f'(3) \) represents the rate at which the number of social - media references to the product is changing 3 days after the product's release.