13. Mariya has purchased a new smart phone and is trying to decide on a service plan. Without a plan each minute of use costs 0.30 and each megabyte of data costs 0.05 A plan that allows unlimited talk and data costs 100/month . Under which circumstances is the plan a better choice for Mariya?

Let 'm' represent the number of minutes Mariya uses per month and 'd' represent the number of megabytes of data she uses per month.<br /><br />The cost without a plan is calculated as: 0.30m + 0.05d<br /><br />The cost with the plan is a fixed $100 per month.<br /><br />The plan is a better choice when the cost without the plan *exceeds* the cost with the plan. So, we're looking for when:<br /><br />0.30m + 0.05d > 100<br /><br />We can express this inequality in different ways to understand the conditions under which the plan is preferable:<br /><br />* **In terms of minutes (given a certain data usage):** m > (100 - 0.05d) / 0.30<br />* **In terms of data (given a certain minute usage):** d > (100 - 0.30m) / 0.05<br /><br />**Example Scenarios:**<br /><br />* **High Data Usage:** If Mariya uses 2500 MB of data per month, the inequality becomes: 0.30m + 0.05 * 2500 > 100 => 0.30m + 125 > 100. In this case, the plan is *always* better regardless of her minute usage, as she'd already be spending more than $100 on data alone.<br /><br />* **High Minute Usage:** If Mariya uses 500 minutes per month, the inequality becomes: 0.30 * 500 + 0.05d > 100 => 150 + 0.05d > 100. Again, the plan is *always* better regardless of her data usage, as she'd already be spending more than $100 on minutes alone.<br /><br />* **Balanced Usage:** If Mariya uses 200 minutes and 1000 MB of data, the cost without a plan is 0.30 * 200 + 0.05 * 1000 = $110. The plan is a better choice in this scenario.<br /><br /><br />**In conclusion:** The plan is a better choice for Mariya if her combined usage of minutes and data results in a cost greater than $100 per month according to the formula 0.30m + 0.05d. High usage of either minutes or data individually can quickly make the plan the more economical option.<br />