Paco's cell phone carrier charges him $0.20 for each text message he sends or receives, $0.15 per minute for calls, and a $15 monthly service fee. Paco is trying to keep his bill for the month below $30. Which best describes the possible values of t, the number of texts he can send or receive?

Question

t can be any real number where 0 ≤ t < 75.
t can be any whole number where 0 ≤ t < 75.
t can be any real number where 0 ≤ t < 150.
t can be any whole number where 0 ≤ t < 150.

GPT-4o mini · Accepted Answer

To determine how many texts Paco can send or receive while keeping his cell phone bill below $30, we set up an inequality based on the costs involved.

Let $ t $ be the number of texts.

The cost for texts is $ 0.20t $.
The cost for calls per minute is $ 0.15m $, where $ m $ is the number of minutes he spends on calls.
There is a fixed monthly service fee of $15.

The total bill can be represented by the equation: \[ \text{Total bill} = 15 + 0.20t + 0.15m \]

We want to keep this total bill below $30: \[ 15 + 0.20t + 0.15m < 30 \]

Subtracting 15 from both sides: \[ 0.20t + 0.15m < 15 \]

Now, to isolate $ t $, we can express $ m $ in terms of $ t $ or analyze it under various scenarios. Assuming that Paco might not make any calls (which gives the maximum limit for $ t $): Let $ m = 0 $: \[ 0.20t < 15 \] Dividing both sides by 0.20: \[ t < 75 \]

While $ t $ can take any value up to 75, considering the context of sending or receiving texts, $ t $ would logically be a whole number (you can’t send a fraction of a text).

Thus, the possible values for $ t $ are any whole numbers where: \[ 0 \leq t < 75. \]

The best description of the possible values of $ t $ is: t can be any whole number where 0 ≤ t < 75.

Answer

t can be any whole number where 0 ≤ t < 75.

GPT-4o mini · Answer

Yes, that's correct!

Given the constraints of Paco's cell phone usage and the monthly bill, the conclusion we reached is that:

t can be any whole number where $ 0 \leq t < 75 $.

This means Paco can send or receive any whole number of texts from 0 up to 74 while keeping his bill below $30, assuming he doesn’t make any calls. If he were to include call minutes, the maximum number of texts he could send or receive would decrease accordingly, but the specified range is accurate for the scenario where no calls are made.

Let me know if you have any further questions or if you need help with anything else!