A cell phone plan has the following charges:
Basic service, including 100 minutes of calls :$20 per month
2nd 100 minutes of calls: $0.075 per minute
Any additional minutes: $0.10 per minute
What is the charge for 200 minutes of calls?
What is the charge for 250 minutes of calls?
Construct a piecewise function that describes the monthly charge C for x minutes of call.
To calculate the charges for different amounts of call minutes, we'll break down the given cell phone plan charges into different tiers.
First, let's find the charges for 200 minutes of calls:
1. The basic service includes 100 minutes of calls for $20 per month.
2. The 2nd 100 minutes of calls are charged at $0.075 per minute.
3. Any additional minutes beyond 200 minutes are charged at $0.10 per minute.
Therefore, the charges for 200 minutes of calls can be calculated as:
Charges = Basic service charges + Charges for 1st and 2nd tier minutes + Charges for additional minutes
1. Basic service charges: $20
2. Charges for 1st and 2nd tier minutes:
- 1st tier (0-100 minutes) = 0 minutes
- 2nd tier (101-200 minutes) = (200 - 100) * $0.075 = $7.50
Total charges for 1st and 2nd tier minutes = $0 + $7.50 = $7.50
3. Charges for additional minutes: (200 - 200) * $0.10 = $0
Total charges for 200 minutes of calls = Basic service charges + Charges for 1st and 2nd tier minutes + Charges for additional minutes
= $20 + $7.50 + $0
= $27.50
Now let's find the charges for 250 minutes of calls:
1. Basic service charges: $20
2. Charges for 1st and 2nd tier minutes:
- 1st tier (0-100 minutes) = 0 minutes
- 2nd tier (101-200 minutes) = 100 * $0.075 = $7.50
Total charges for 1st and 2nd tier minutes = $0 + $7.50 = $7.50
3. Charges for additional minutes:
- Additional minutes (201-250 minutes) = (250 - 200) * $0.10 = $5.00
Total charges for 250 minutes of calls = Basic service charges + Charges for 1st and 2nd tier minutes + Charges for additional minutes
= $20 + $7.50 + $5.00
= $32.50
Now, let's construct the piecewise function that describes the monthly charge C for x minutes of call:
C(x) = $20, for x ≤ 100
C(x) = $20 + ($0.075 * (x - 100)), for 100 < x ≤ 200
C(x) = $20 + ($0.075 * 100) + ($0.10 * (x - 200)), for x > 200
Note: This piecewise function assumes that the charges for 1st and 2nd tier minutes are cumulative.