Explanation
The formula of a linear function in slope-intercept form looks like this:
[tex]f(x)=mx+b[/tex]
Where m is the slope and b is the y-intercept.
We must model the profits of an artist using this equation.
1. We are told that the distributor charges the artist $50. This means that there's a flat value of -50 in the function i.e. the y-intercept is -50 and we have b=-50.
2. The slope of the function tells us how much does the function increases when x increases in 1 unit. Our function gives us the amount of profits per total number of streams which means that the payment per stream is equal to its slope. Therefore the slope is 5.
3. Now that we have the slope m and the y-intercept b we can write the equation of the line in slope-intercept form (replacing f(x) with y): y = 5x - 50
4. With the equation of the line and the values of m and b you can graph the function, it should look like this:
5. We must find the number of streams where the distributor charges are paid. If this is the break even point then it means that for this value of x the profits are 0 i.e. y=0. Then we must solve the equation given by taking y=0:
[tex]0=5x-50[/tex]
We add 50 to both sides and then we divide both sides by 5:
[tex]\begin{gathered} 0+50=5x-50+50 \\ 50=5x \\ \frac{50}{5}=\frac{5x}{5} \\ x=10 \end{gathered}[/tex]
So the answer to part 5 is 10 streams.
6. Here we basicaly have to find the value of x (the number of streams) that makes y (the profits) equal to 950,000. Then we just need to take y=950,000 and solve for x:
[tex]950,000=5x-50[/tex]
We add 50 to both sides we obtain:
[tex]\begin{gathered} 950,000+50=5x-50+50 \\ 5x=950,050 \end{gathered}[/tex]
Then we divide by 5:
[tex]\begin{gathered} \frac{5x}{5}=\frac{950,050}{5} \\ x=190,010 \end{gathered}[/tex]
So it will take 190,010 streams to profit $950,000.
7. Here we just need to take x=9.1 billions in the equation. Then we have:
[tex]y=5\cdot9,100,000,000-50=45,499,999,950[/tex]
So he may have made $45,499,999,950.
