A phone plan costs 30 dollars per month plus 0.05 dollars per minute of talk and 0.10 dollars per text sent (received texts are free). Write an expression for the monthly cost using t minutes and x texts, then find the cost for (t, x) = (100, 50), (124, 26), (75, 75), (80, 95) and (50, 100).

Step-by-step solution

1. 1

   (a) Identify the fixed part and the variable parts. The \30$ is charged whether or not the phone is used, so it is a constant term. The other two charges are rates — dollars per unit — so each becomes a coefficient times its variable:

   $C=30+0.05t+0.10x$

   Received texts contribute nothing, which is why only sent texts appear.

2. 2

   (b) Evaluate the first two rows. Substitute and keep the two rate terms separate:

   $(100,50):\ 30+0.05(100)+0.10(50)=30+5+5=\$40.00$

   $(124,26):\ 30+0.05(124)+0.10(26)=30+6.20+2.60=\$38.80$

3. 3

   Evaluate the remaining three rows.

   $(75,75):\ 30+3.75+7.50=\$41.25$

   $(80,95):\ 30+4.00+9.50=\$43.50$

   $(50,100):\ 30+2.50+10.00=\$42.50$

4. 4

   Collect the table.

   Talk (min)	Texts sent	Total cost
   100	50	$40.00
   124	26	$38.80
   75	75	$41.25
   80	95	$43.50
   50	100	$42.50

5. 5

   (c) Compare the two rates directly. A text costs \0.10$and a minute costs$$0.05$, so one text costs the same as two minutes of talk. For the same number of extra units it is therefore cheaper to talk more and text less.

   $\boxed{C=30+0.05t+0.10x;\ \text{talking is half the price of texting per unit}}$

6. 6

   Check the conclusion against the table. The cheapest row, \38.80$, is the one with the fewest texts ($26$), even though it has the most talk minutes ($124$). The most expensive,$$43.50$, has$95$texts. That ordering matches the rate comparison rather than the total number of units used —$(124,26)$uses$150$units but costs less than$(50,100)$, which uses the same$150$ units.