Question

1. on average, susan downloads 60 songs per month. an online music vendor sells package prices for songs that can be downloaded onto personal digital devices. the graph below shows the package prices for the most popular promotions. susan wants to know if she should buy her music from this company or pay a flat fee of $58.00 per month offered by another company. which is the better buy? a. find the constant of proportionality for this situation. b. write an equation to represent the relationship. c. use your equation to find the answer to susans question above. justify your answer with mathematical evidence and a written explanation.

Explanation:

Step1: Find constant of proportionality

The constant of proportionality $k$ in a proportional relationship $y = kx$ (where $y$ is total cost and $x$ is number of songs) can be found using any point $(x,y)$ on the line. Using the point $(5,4.5)$, we have $k=\frac{y}{x}=\frac{4.5}{5}= 0.9$.

Step2: Write the equation

The equation representing the relationship between the total cost $y$ and the number of songs $x$ is $y = 0.9x$ since the constant of proportionality $k = 0.9$.

Step3: Calculate cost for 60 songs

If Susan downloads 60 songs per month, using the equation $y=0.9x$, when $x = 60$, we have $y=0.9\times60=54$.

Step4: Compare costs

The flat - rate company charges $58$ per month. Since $54<58$, it is better for Susan to buy music from the online music vendor.

Answer:

a. The constant of proportionality is $0.9$.
b. The equation is $y = 0.9x$.
c. It is better for Susan to buy music from the online music vendor. When Susan downloads 60 songs, the cost from the online music vendor is $y=0.9\times60 = 54$ dollars, while the flat - rate company charges $58$ dollars per month. Since $54<58$, the online music vendor is the better buy.