Kim Davis has decided to purchase a cellular phone, but she is ensure about which rate plan to select. The regular plan charges a fixed fee of $55 per month for 1,000 minutes of airtime plus $0.33 per minute for any time over 1,000 minutes. The executive plan charges a fixed fee of $100 per month for 1,200 minutes airtime plus $0.25 per minute over 1,200 minutes. If Kim expects to use the phone for 21 hours per month, which plan should she select? At what level of use would Kim be indifferent between the two plans?

In other words, the executive plan is cheaper.

562.5 minutes, any plan is indifferent to it.

Step-by-step explanation:

We have to Kim will use the phone 21 hours a month, that in minutes would be equal to:

21 h * (60 m / 1 h) = 1260 minutes

Now, let's pose each case.

Regular plan, which give away a total of 1000 minutes, therefore Kim must pay extra charge 1260 - 1000 = 260. Now we have to:

$ X = Fixed charge + Extra charges

We know that the fixed charge is $ 55 and that for each extra minute it is $ 0.33, replacing

$ X = 55 + 260 * 0.33 = $ 140.8

Executive plan, which give away a total of 1200 minutes, therefore Kim must pay extra charge 1260 - 1200 = 60. Now we have to:

$ Y = Fixed charge + Extra charges

We know that the fixed charge is $ 100 and that for each extra minute it is $ 0.25, replacing

$ Y = 100 + 60 * 0.25 = $ 115

In other words, the executive plan is cheaper.

It is indifferent when X and Y are equal, we calculate the number of minutes (m) when these values are equal. So:

55 + m * 0.33 = 100 + m * 0.25

0.33 * m - 0.25 * m = 100 - 55

0.08 * m = 45

m = 45 / 0.08

m = 562.5

That is, when in both cases, the number of extra minutes that must be paid equals 562.5, any plan is indifferent to it.

She should select the second plan, since she'd pay 125 $ for that instead of 140.8 $ for the first one.

To be indifferent between the plans she'd have to use 2312.5 minutes per month.

Step-by-step explanation:

First plan:

value to pay = 55 + variable amount

if minutes used are greater than 1000:

variable amount = (minutes used - 1000) * 0.33

Second plan:

value to pay = 100 + variable amount

if minutes used are greater than 1200:

variable amount = (minutes used - 1200) * 0.25

If she wants to use her phone for 21 h per month that is the same as 21*60 = 1260 minutes per month so:

Plan 1: value to pay = 55 + (1260-1000) * 0.33 = 55 + 85.8 = 140.8 $

Plan2: value to pay = 100 + (1260-1200) * 0.25 = 60 + 65 = 125 $

In this case she should select the first plan.

To compute the value both plans are equal we need to equate both expressions, so we have:

55 + (minutes used - 1000) * 0.33 = 100 + (minutes used - 1200) * 0.25

55 + 0.33 * (minutes used) - 330 = 100 + 0.25 * (minutes used) - 300

0.33 * (minutes used) - 0.25 * (minutes used) = 55 + 330 + 100 - 300

0.08 * (minutes used) = 185

minutes used = 2312.5

She would have to use 2312.5 minutes in order to be indifferent between the plans.