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Recall from a previous lesson 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 $y=5.25+0.30 x$.



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 $y=0.35 x+4.50$.



Explain the meaning of the value 4.50 in the new equation. Is this a better situation for Kelly than before?

Verified Answer :

  • The constant term \(4.50\) represents the fixed monthly subscription fee Kelly pays.
  • The break-even point between the two models occurs at 15 songs, where both models result in a total cost of \(\$9.75\).
  • The new model is more cost-effective for Kelly if she downloads fewer than 15 songs per month.
  • Final Answer: The value \(4.50\) is the fixed monthly fee. The new plan is better if she downloads fewer than 15 songs \(\boxed{x < 15}\).

Explanation

  1. Meaning of the Constant Term

To understand the pricing structure for Kelly, we analyze the linear cost function \(y = mx + b\), where \(y\) represents the total monthly cost, \(x\) is the number of songs downloaded, \(m\) is the cost per song, and \(b\) is the fixed monthly subscription fee (the y-intercept). In the new model, \(y = 0.35x + 4.50\), the constant term \(4.50\) represents the fixed monthly subscription fee. This means Kelly must pay \(\$4.50\) every month regardless of how many songs she downloads.

  1. Comparing the Two Models

To determine if this is a better situation, we compare the new cost to the original cost. The old cost was \(y = 0.30x + 5.25\). We look for the break-even point where the cost of both models is identical:
\[0.35x + 4.50 = 0.30x + 5.25\]
Subtracting \(0.30x\) from both sides gives:
\[0.05x + 4.50 = 5.25\]
Subtracting \(4.50\) from both sides:
\[0.05x = 0.75\]
Dividing by \(0.05\):
\[x = \frac{0.75}{0.05} = 15\]
The costs are equal when Kelly downloads exactly 15 songs per month.

  1. Determining the Better Option

Whether the new plan is better for Kelly depends on how many songs she downloads:

  • If Kelly downloads fewer than 15 songs per month (\(x < 15\)), the new model is cheaper because the lower fixed fee (\(\$4.50\) vs \(\$5.25\)) outweighs the higher cost per song.
  • If she downloads exactly 15 songs, the cost is the same (\(\$9.75\)).
  • If she downloads more than 15 songs (\(x > 15\)), the old model was cheaper because the lower cost per song (\(\$0.30\) vs \(\$0.35\)) becomes more significant.

Thus, the new situation is better if she downloads fewer than 15 songs.

Examples

Linear cost modeling is a fundamental tool for businesses and consumers alike. For instance, consider choosing between different mobile data plans: one might have a high monthly fee but cheap per-gigabyte costs, while another has a low monthly fee but expensive data overages. By setting these models equal to each other, you can calculate the exact usage threshold where one plan becomes more cost-effective than the other, allowing for smart financial planning.

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