Question

EMERGENCY! 50 PTS

For the month of June, a music supply company is promoting the launch of its electronic music club by offering new members two options for downloading music.
Option #1: Annual club membership at a fee of $125 per year plus an option of prepaying for unlimited music downloads at a rate of $10 per month.
Option #2: Annual club membership at a fee of $50 per year plus an option of prepaying for music downloads at a rate of $1 per song.
You decide to join the electronic music club for one year, but aren’t sure which membership option is best suited for your music downloading needs. Review both options that the music supply company is offering for its new club members. Use the information provided to respond to the following prompts. When necessary, answer in complete sentences and include all calculations.
Write a function that best models the total cost of club membership plus downloads under option #1.
For membership option #1, calculate the total cost for one year of club membership and prepaid unlimited music downloads.
Write a function that best models the total cost of club membership plus downloads under option #2
Calculate the maximum number of music downloads that you will have with one year of club membership under option #2 for the same annual cost of a membership with unlimited downloads.
Over the past year, you downloaded an average of 14 songs per month. Assuming that for the next year, you continue to download music at the same rate per month, which membership option is most cost effective?

Answer 1

The best option depends on how many songs you plan to purchase.

Option 1 will be the best option if you wish to purchase an unlimited amount of songs or more than 16 per month for the total cost of one year of $245. It's modeled by the equation y=10x+125 where x is months and y is total annual cost.

Option 2 will be the best option if you wish to purchase less than 16 songs a month. This would be 16(12) = 192 songs a year. To stay under the cost of option 1 you can buy only 195 songs in a year. Paying for each song individually at $1 plus a lower annual fee will result in you spending $242 dollars a year or less if you purchase less songs. It's modeled by the equation y=1s+50 where s is the number of songs downloaded in a year and y is total annual cost.

Because your average of 14 songs a month is less than 16 songs a month, option 2 is your best option. You will download all the songs you normally do and pay less than option 1. Your cost under Option 1 would be $245. Your cost under Option 2 would be $218.

Explanation:

Each option in the electronic music club represents a linear function. Each option has a constant rate of change charged per month or song. This is your slope. Each option has a one time annual fee charged. This is your y-intercept.

Option 1 has a constant rate (m) of $10 per month with an annual fee (b) of $125. Using Slope-Intercept form, y=mx+b, we can write the function y=10x+125 where x is number of months and y is total cost in 12 months.

Option 2 has a constant rate (s) of $1 per song with an annual fee (b) of $50. Using Slope-Intercept form, y=mx+b, we can write the function y=1s+50 where s is number of songs and y is total cost in 12 months.

We know the total annual cost of Option 1 by substituting x=12 months to represent a year.

y=10(12)+125=120+125=245.

We find when Option 2 will be the same cost by substituting y=245 into y=1s+50 and solve for s.

245=1s+50

245-50=1s+50-50

195=1s

This means that both options will be the same cost for one year if we purchase 195 songs that year. If we purchase more, than Option 1 is the best choice. If we purchase less than 195, Option 2 remains the best choice.

Lastly, if your average number of songs is 14 per month, we can find 14(12)=168 songs in a year. This is lower than 195. Thus option 2 is the best answer for you.