A utility company has the following rate schedule for natural gas usage in a city’s business district:
Monthly service charge $8.80
Per therm service charge:
1st 25 therms : $0.6686/therm
Over 25 therms: $0.85870/therm
(a) What is the charge for using 25 therms in one month?
(b) What is the charge for using 45 therms in one month?
(c) Construct a function that gives the monthly charge C for x therms of gas.
*Please show work
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(a) To find the charge for using 25 therms in one month, we need to calculate the cost taking into account the per therm service charge.
The first 25 therms have a service charge of $0.6686 per therm.
Charge for using 25 therms = (25 therms) * ($0.6686/therm) = $16.72
Therefore, the charge for using 25 therms in one month is $16.72.
(b) To find the charge for using 45 therms in one month, we need to calculate the cost taking into account the per therm service charge.
The first 25 therms have a service charge of $0.6686 per therm, and any therms beyond that have a service charge of $0.85870 per therm.
For the first 25 therms: Charge = (25 therms) * ($0.6686/therm) = $16.72
For the additional 20 therms: Charge = (20 therms) * ($0.85870/therm) = $17.17
Total charge for using 45 therms = Charge for the first 25 therms + Charge for the additional 20 therms = $16.72 + $17.17 = $33.89
Therefore, the charge for using 45 therms in one month is $33.89.
(c) To construct a function that gives the monthly charge C for x therms of gas, we need to take into account the different service charges based on the number of therms used.
For the first 25 therms, the service charge is $0.6686 per therm.
For any therms beyond 25, the service charge is $0.85870 per therm.
If x is the number of therms,
C(x) = $8.80 (monthly service charge) + ($0.6686 per therm) * x (for the first 25 therms) + ($0.85870 per therm) * (x-25) (for therms beyond 25)
Therefore, the function that gives the monthly charge C for x therms of gas is:
C(x) = $8.80 + ($0.6686 * x), if x <= 25
C(x) = $8.80 + ($0.6686 * 25) + ($0.85870 * (x-25)), if x > 25
To find the charges for using a certain number of therms in one month, we need to calculate the charges according to the rate schedule provided.
(a) To find the charge for using 25 therms in one month:
The rate for the first 25 therms is $0.6686/therm.
So, the charge for using 25 therms is:
25 therms × $0.6686/therm = $16.715
(b) To find the charge for using 45 therms in one month:
For the first 25 therms, the rate is $0.6686/therm.
For the remaining 45 - 25 = 20 therms, the rate is $0.85870/therm.
So, the charge for using 45 therms is:
25 therms × $0.6686/therm + 20 therms × $0.85870/therm
= $16.715 + $17.174 = $33.889
(c) The function that gives the monthly charge C for x therms of gas can be constructed using the rate schedule:
Let's define C(x) as the monthly charge for x therms.
The charge consists of two parts:
1. The fixed monthly service charge of $8.80.
2. The variable charge for the therms used.
If x ≤ 25, then the variable charge is calculated as x therms × $0.6686/therm.
If x > 25, then the charge for the first 25 therms is $0.6686/therm, and for the remaining therms (x - 25), the charge is ($0.85870/therm) × (x - 25).
So, the function C(x) can be defined as:
C(x) = $8.80 + { $0.6686/therm × x if x ≤ 25
{ $0.6686/therm × 25 + ($0.85870/therm) × (x - 25) if x > 25
For example, to find the charge for using 45 therms, we can substitute x = 45 into the function:
C(45) = $8.80 + { $0.6686/therm × 45 if 45 ≤ 25
{ $0.6686/therm × 25 + ($0.85870/therm) × (45 - 25) if 45 > 25
Simplifying this expression will give us the numerical charge.