Questions: The Trussville Utilities uses the rates shown in the table below to compute the monthly cost, C(x), of natural gas for residential customers. Usage, x, is measured in cubic hundred feet (CCF) of natural gas. Base charge 7.00 First 1000 CCF 0.05 per CCF Over 1000 CCF 0.10 per CCF a. Find the charge for using 600 CCF. b. Find an expression for the cost function C(x) for usage under 1000 CCF. C(x)= c. Find an expression for the cost function C(x) for usage over 1000 CCF. C(x)=

Solution Steps

To solve the given problem, we need to compute the cost of natural gas usage based on the provided rates. We will break down the problem into three parts:

a. Calculate the charge for using 600 CCF. b. Derive an expression for the cost function \( C(x) \) for usage under 1000 CCF. c. Derive an expression for the cost function \( C(x) \) for usage over 1000 CCF.

1. For part (a), we will use the base charge and the rate for the first 1000 CCF to calculate the cost for 600 CCF.
2. For part (b), we will create a cost function \( C(x) \) for \( x \leq 1000 \) CCF, which includes the base charge and the rate for the first 1000 CCF.
3. For part (c), we will create a cost function \( C(x) \) for \( x > 1000 \) CCF, which includes the base charge, the rate for the first 1000 CCF, and the rate for usage over 1000 CCF.

To find the charge for using 600 CCF, we use the base charge and the rate for the first 1000 CCF: \[ \text{Base charge} = \$7.00 \] \[ \text{Rate for first 1000 CCF} = \$0.05 \text{ per CCF} \] \[ \text{Usage} = 600 \text{ CCF} \] \[ \text{Cost} = 7.00 + (0.05 \times 600) = 7.00 + 30.00 = 37.00 \] \[ \boxed{\$37.00} \]

For usage \( x \leq 1000 \) CCF, the cost function includes the base charge and the rate for the first 1000 CCF: \[ C(x) = 7.00 + 0.05x \quad \text{for} \quad x \leq 1000 \] \[ \boxed{C(x) = 7.00 + 0.05x} \]

For usage \( x > 1000 \) CCF, the cost function includes the base charge, the rate for the first 1000 CCF, and the rate for usage over 1000 CCF: \[ \text{Base charge} = \$7.00 \] \[ \text{Rate for first 1000 CCF} = \$0.05 \text{ per CCF} \] \[ \text{Rate for usage over 1000 CCF} = \$0.10 \text{ per CCF} \] \[ C(x) = 7.00 + (0.05 \times 1000) + 0.10(x - 1000) \quad \text{for} \quad x > 1000 \] \[ C(x) = 7.00 + 50.00 + 0.10(x - 1000) \] \[ C(x) = 57.00 + 0.10(x - 1000) \] \[ \boxed{C(x) = 57.00 + 0.10(x - 1000)} \]

Final Answer