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Recall that Kelly wants to add new music to her MP3 player. She was interested in a monthly subscription site that offered its MP3 downloading service for a monthly subscription fee plus a fee per song. The linear function that modeled the total monthly cost in dollars ($y$) based on the number of songs downloaded ($x$) is $y=5.25+0.30x$.



The site has suddenly changed its monthly price structure. The linear function that models the new total monthly cost in dollars ($y$) based on the number of songs downloaded ($x$) is $y=0.35x+4.50$.


AExplain the meaning of the value 4.50 in the new equation. Is this a better situation for Kelly than before?
BExplain the meaning of the value 0.35 in the new equation. Is this a better situation for Kelly than before?
CIf you were to graph the two equations (old versus new), which line would have the steeper slope? What does this mean in the context of the problem?
DWhich subscription plan provides the better value if Kelly downloads fewer than 15 songs per month?

Verified Answer :

  • The fixed monthly fee is \(\$4.50\) (better than \(\$5.25\)), while the per-song cost is \(\$0.35\) (worse than \(\$0.30\)).
  • The new line is steeper, meaning the total cost grows faster as more songs are downloaded.
  • Setting the two equations equal confirms they cross at \(x = 15\).
  • For fewer than \(15\) songs, the new plan is more economical because the lower fixed fee outweighs the higher per-song cost.
  • Final answer: The new plan is better for \(x < 15\).

Explanation

  1. Data and Problem Analysis

The old monthly cost function is represented by \(y_{old} = 0.30x + 5.25\), where \(x\) is the number of songs downloaded and \(y\) is the total monthly cost in dollars. The new monthly cost function is given by \(y_{new} = 0.35x + 4.50\). In both linear functions of the form \(y = mx + b\), the variable \(m\) (the slope) represents the cost per individual song, and the variable \(b\) (the y-intercept) represents the fixed monthly subscription fee. To answer the questions, we need to analyze these components and compare the two pricing structures.

  1. Evaluating Cost Components
AThe value \(4.50\) in the new equation represents the fixed monthly subscription fee. Compared to the old fixed fee of \(\$5.25\), this is a lower cost, which makes this aspect of the new situation better for Kelly.
BThe value \(0.35\) in the new equation represents the variable cost per song downloaded, meaning it costs \(\$0.35\) for every new song Kelly downloads. Comparing this to the old rate of \(\$0.30\) per song, the cost per song has increased by \(\$0.05\). Therefore, this aspect of the new situation is not as good for Kelly as it was before.
  1. Comparing Plans and Determining Value
CThe slope of the new line is \(0.35\), while the slope of the old line is \(0.30\). Since \(0.35 > 0.30\), the new line is steeper than the old one. In the context of the problem, a steeper slope means that the total cost increases more rapidly with each additional song downloaded under the new plan compared to the old one.
DTo find which plan is better for \(x < 15\), we compare the two functions. The lines intersect where \(0.30x + 5.25 = 0.35x + 4.50\). Solving this for \(x\):

\[0.75 = 0.05x \implies x = 15.\]
For any number of songs \(x < 15\), the cost of the new plan will be lower. For example, if Kelly downloads \(10\) songs:

  • Old plan: \(0.30(10) + 5.25 = \$8.25\)
  • New plan: \(0.35(10) + 4.50 = \$8.00\)

Since \(\$8.00 < \$8.25\), the new subscription plan is the better value for Kelly when downloading fewer than \(15\) songs per month.

Examples

This type of analysis is crucial in personal finance and business. For example, when choosing between two cell phone plans, one might have a high monthly fee but cheap per-minute rates, while another has a low monthly fee but expensive per-minute rates. By modeling these costs as linear functions \(y = mx + b\), consumers can determine exactly how many minutes they need to use each month to decide which plan is more cost-effective. These algebraic comparisons help in making informed decisions about spending and resource allocation.

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