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
To find the charge for using a certain number of therms in one month, you need to calculate the total cost based on the rate schedule provided by the utility company.
(a) The charge for using 25 therms in one month can be calculated as follows:
- Monthly service charge: $8.80
- 1st 25 therms: $0.6686/therm
First, multiply the rate per therm by the number of therms to get the cost for the 1st 25 therms:
25 therms * $0.6686/therm = $16.715
Then, add the monthly service charge to get the total charge:
$16.715 + $8.80 = $25.515
Therefore, the charge for using 25 therms in one month is $25.515.
(b) The charge for using 45 therms in one month can be calculated in a similar way. However, since the number of therms exceeds the 1st 25 therms, we need to consider both the cost for the 1st 25 therms and the cost for the additional therms.
- Monthly service charge: $8.80
- 1st 25 therms: $0.6686/therm
- Over 25 therms: $0.85870/therm
First, calculate the cost for the 1st 25 therms:
25 therms * $0.6686/therm = $16.715
Next, calculate the cost for the additional therms (45 - 25 = 20 therms):
20 therms * $0.85870/therm = $17.174
Then, add the monthly service charge, cost for the 1st 25 therms, and cost for the additional therms:
$8.80 + $16.715 + $17.174 = $42.689
Therefore, the charge for using 45 therms in one month is $42.689.
(c) The function that gives the monthly charge c for x therms of gas can be expressed as follows:
c(x) = $8.80 + ($0.6686 * min(x, 25)) + ($0.85870 * max(0, (x - 25)))
In this function, min(x, 25) represents the minimum value between x and 25, and max(0, (x - 25)) represents the maximum value between 0 and (x - 25). This function considers the cost for the 1st 25 therms at the rate $0.6686/therm and the cost for any additional therms at the rate $0.85870/therm, while also including the monthly service charge.