To solve this problem, let's break it down step-by-step.
We know that the average number of weekly social media posts ([tex]\( P \)[/tex]) varies jointly with the poster's yearly income ([tex]\( I \)[/tex]) in thousands and inversely with their age ([tex]\( A \)[/tex]) in years. This can be expressed mathematically as:
[tex]\[
P = k \times \frac{I}{A}
\][/tex]
where:
- [tex]\( P \)[/tex] is the number of posts,
- [tex]\( I \)[/tex] is the yearly income in thousands,
- [tex]\( A \)[/tex] is the age,
- [tex]\( k \)[/tex] is the proportionality constant we need to find.
Given:
- [tex]\( P = 300 \)[/tex] (the average number of weekly posts),
- [tex]\( A = 60 \)[/tex] years,
- [tex]\( I = 100 \)[/tex] (since the income is \$100,000 and we consider it in thousands, so [tex]\( 100,000/1000 = 100 \)[/tex]).
We substitute these values into the equation to solve for [tex]\( k \)[/tex]:
[tex]\[
300 = k \times \frac{100}{60}
\][/tex]
Simplify the fraction [tex]\(\frac{100}{60}\)[/tex]:
[tex]\[
\frac{100}{60} = \frac{10}{6} = \frac{5}{3}
\][/tex]
So, the equation becomes:
[tex]\[
300 = k \times \frac{5}{3}
\][/tex]
To isolate [tex]\( k \)[/tex], multiply both sides of the equation by [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[
300 \times \frac{3}{5} = k
\][/tex]
[tex]\[
k = 180
\][/tex]
Thus, the value of [tex]\( k \)[/tex] is 180. The correct answer is:
[tex]\[
k = 180
\][/tex]
