Kim Davis has decided to purchase a cellular phone, but she is unsure 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 of airtime plus $.025 per minute over 1,200 minutes.

a. If Kim expects to use the phone for 21 hours per month, which plan should she select?
b. At what level of use would Kim be indifferent between the two plans? Hint: try to set the two plans equal.

937.5

a. Well, if Kim expects to use the phone for 21 hours per month, let's calculate the minutes she'll be using. Since there are 60 minutes in an hour, Kim would use 21 hours * 60 minutes = 1260 minutes.

For the "regular" plan, Kim would pay a fixed fee of $55 plus 260 minutes * $0.33 per minute. So her total cost would be $55 + (260 * $0.33) = $55 + $85.80 = $140.80.

For the "executive" plan, Kim would pay a fixed fee of $100 plus 60 minutes * $0.025 per minute. So her total cost would be $100 + (60 * $0.025) = $100 + $1.50 = $101.50.

Therefore, Kim should select the "executive" plan as it would be cheaper for her at $101.50 compared to $140.80 for the "regular" plan.

b. To find the level of use where Kim would be indifferent between the two plans, we can set their costs equal to each other.

Let's denote the number of minutes used beyond the inclusive minutes as 'x'.

For the "regular" plan, the cost would be $55 + (x * $0.33).

For the "executive" plan, the cost would be $100 + (x * $0.025).

Setting these two costs equal, we have:

$55 + (x * $0.33) = $100 + (x * $0.025).

Now, let's solve for 'x':

$0.33x - $0.025x = $100 - $55.

Simplifying:

$0.305x = $45.

Dividing both sides by $0.305:

x = $45 / $0.305 = 147.54 minutes.

Therefore, Kim would be indifferent between the two plans at around 148 minutes of usage beyond the inclusive minutes.

a. To determine which plan Kim should select, we can calculate the cost for each plan based on her expected usage.

For the regular plan:
- Fixed fee: $55
- Airtime included: 1,000 minutes
- Cost per additional minute: $0.33

Kim expects to use the phone for 21 hours per month, which is equal to 21 * 60 = 1260 minutes. Since this is over 1,000 minutes, she will have to pay for the additional minutes.

Additional minutes = 1260 - 1000 = 260 minutes
Cost for additional minutes = 260 * $0.33 = $85.80

Total cost for the regular plan = Fixed fee + Cost for additional minutes = $55 + $85.80 = $140.80

For the executive plan:
- Fixed fee: $100
- Airtime included: 1,200 minutes
- Cost per additional minute: $0.025

Kim's expected usage of 1260 minutes is still below the 1,200 minutes included in the executive plan, so she will not have to pay for any additional minutes.

Total cost for the executive plan = Fixed fee = $100

Comparing the two plans, the regular plan would cost Kim $140.80, while the executive plan would cost her $100. Therefore, Kim should select the executive plan since it would be cheaper for her.

b. To find the level of use at which Kim would be indifferent between the two plans, we can set the total costs of the regular and executive plans equal to each other.

Let x be the number of additional minutes used.

For the regular plan:
Total cost = Fixed fee + Cost for additional minutes
Total cost = $55 + $0.33x

For the executive plan:
Total cost = Fixed fee
Total cost = $100

Setting these two equations equal to each other:
$55 + $0.33x = $100

Simplifying:
$0.33x = $45
x = $45 / $0.33
x ≈ 136.36

Therefore, Kim would be indifferent between the two plans when she uses approximately 136 additional minutes.

To determine which rate plan Kim should select, we need to compare the costs of the regular plan and the executive plan based on her expected usage.

a. For the regular plan:
- The fixed fee is $55 per month.
- The plan includes 1,000 minutes of airtime.
- Additional minutes over 1,000 are charged at a rate of $0.33 per minute.

To calculate the cost of the regular plan, we need to find out how many minutes Kim expects to use in total. We know that she plans to use the phone for 21 hours per month, but we need to convert this to minutes.

Since there are 60 minutes in an hour, 21 hours is equal to 21 x 60 = 1,260 minutes.

Now, let's calculate the cost of the regular plan based on Kim's expected usage of 1,260 minutes:
- The first 1,000 minutes are covered by the fixed fee, so there is no additional charge for those.
- The remaining 1,260 - 1,000 = 260 minutes will be charged at $0.33 per minute.
- The cost for these additional minutes is 260 x $0.33 = $85.8.

Therefore, the total cost for the regular plan would be $55 (fixed fee) + $85.8 (additional minutes) = $140.8.

b. For the executive plan:
- The fixed fee is $100 per month.
- The plan includes 1,200 minutes of airtime.
- Additional minutes over 1,200 are charged at a rate of $0.025 per minute.

Again, we need to calculate the cost of the executive plan based on Kim's expected usage of 1,260 minutes:
- The first 1,200 minutes are covered by the fixed fee, so there is no additional charge for those.
- The remaining 1,260 - 1,200 = 60 minutes will be charged at $0.025 per minute.
- The cost for these additional minutes is 60 x $0.025 = $1.5.

Thus, the total cost for the executive plan would be $100 (fixed fee) + $1.5 (additional minutes) = $101.5.

Comparing the two plans:
- The regular plan would cost $140.8.
- The executive plan would cost $101.5.

Based on Kim's expected usage of 1,260 minutes per month, she should select the executive plan as it would be cheaper for her.

To find the level of use at which Kim would be indifferent between the two plans, we can set the costs of the regular and executive plans equal to each other:

$55 + (minutes used - 1,000) x $0.33 = $100 + (minutes used - 1,200) x $0.025

Simplifying the equation gives:

0.33 x minutes used - 330 = 0.025 x minutes used - 30

0.33 x minutes used - 0.025 x minutes used = 330 - 30

0.305 x minutes used = 300

minutes used = 300 / 0.305

minutes used ≈ 983.61

Therefore, Kim would be indifferent between the two plans when she uses approximately 984 minutes.