A hotel chain charges $75 each night for the first two nights and $50 for each additional night’s stay the total cost function T is a function of the number of nights x that a guest stays write the piecewise function T(x) and then calculate T(2) T(3) T(5)
We know that the hotel charges $75 for the first two nights. Thus, for x = 1 and x = 2, the cost is $75 each night. For x = 3 and any subsequent night, the cost is $50 each night.
Therefore, we can write the piecewise function T(x) as:
T(x) =
75 if x = 1 or x = 2,
50(x-2) if x ≥ 3.
Now let's calculate T(2), T(3), and T(5):
T(2) = 75 (as per the first condition of the piecewise function)
T(3) = 50(3-2) = 50 (as per the second condition of the piecewise function)
T(5) = 50(5-2) = 150 (as per the second condition of the piecewise function)
To write the piecewise function T(x), we need to consider two scenarios: when the guest stays for only two nights, and when the guest stays for more than two nights.
When the guest stays for only two nights, the cost is $75 per night. Therefore, the function can be written as:
T(x) = 75x if x ≤ 2
When the guest stays for more than two nights, the cost is $75 for the first two nights and $50 for each additional night. This means the guest pays $75 for the first two nights and $50 per night for x - 2 additional nights. Therefore, the function can be written as:
T(x) = 75(2) + 50(x - 2) if x > 2
Now, let's calculate T(2), T(3), and T(5):
1. T(2):
Since x = 2, we use the first part of the function:
T(2) = 75(2) = 150
2. T(3):
Since x > 2, we use the second part of the function:
T(3) = 75(2) + 50(3 - 2) = 150 + 50 = 200
3. T(5):
Since x > 2, we still use the second part of the function:
T(5) = 75(2) + 50(5 - 2) = 150 + 150 = 300
Therefore, T(2) = 150, T(3) = 200, and T(5) = 300.