Westside Energy charges its electric customers a base rate of $4.00 per month, plus 10¢ per kilowatt-hour (KWh) for the first 300 kWh used and 26¢ per kWh for all usage over 300 kWh. Suppose a customer uses \(x\) kWh of electricity in one month.
(a) Express the monthly cost \(E\) as a piecewise defined function of \(x\). (Assume \(E\) is measured in dollars.)
\(E(x)=\left\{\begin{array}{ll} -1 / 2 & text { if } 0 \leq x \leq 300 \\ -1 / 2 & text { if } 300
Answer (2)
For 0 ≤ x ≤ 300 , the cost is the base rate plus $0.10 per kWh: E ( x ) = 4 + 0.10 x .
For 300"> x > 300 , the cost is the base rate plus $0.10 for the first 300 kWh and $0.26 for the rest: E ( x ) = 4 + 0.10 ( 300 ) + 0.26 ( x − 300 ) .
Simplify the expression for 300"> x > 300 : E ( x ) = 0.26 x − 44 .
The piecewise function is: 300 \end{cases}}"> E ( x ) = { 4 + 0.10 x if 0 ≤ x ≤ 300 0.26 x − 44 if x > 300
Explanation
Understanding the Problem We want to express the monthly cost E as a piecewise defined function of x , where x is the number of kilowatt-hours (kWh) used in one month. The base rate is $4.00 per month, the cost for the first 300 kWh is $0.10 per kWh, and the cost for usage over 300 kWh is $0.26 per kWh.
Cost for 0 to 300 kWh If 0 ≤ x ≤ 300 , the monthly cost E is the base rate plus the cost for the first x kWh at $0.10 per kWh. So, we have: E ( x ) = 4 + 0.10 x
Cost for over 300 kWh If 300"> x > 300 , the monthly cost E is the base rate plus the cost for the first 300 kWh at $0.10 per kWh plus the cost for the remaining ( x − 300 ) kWh at $0.26 per kWh. So, we have: E ( x ) = 4 + 0.10 ( 300 ) + 0.26 ( x − 300 ) Simplifying the expression: E ( x ) = 4 + 30 + 0.26 x − 78 = 0.26 x − 44
Final Answer Therefore, the piecewise function is: 300 \end{cases}"> E ( x ) = { 4 + 0.10 x if 0 ≤ x ≤ 300 0.26 x − 44 if x > 300 So the final answer is: 300 \end{cases}"> E ( x ) = { 4 + 0.10 x if 0 ≤ x ≤ 300 0.26 x − 44 if x > 300
Examples
Piecewise functions are useful in real life for modeling situations where different rules or conditions apply over different intervals. For example, cell phone plans often have a fixed monthly fee for a certain amount of data usage, and then charge a different rate for additional data used beyond that limit. Similarly, income tax brackets are structured as piecewise functions, where different tax rates apply to different ranges of income. Understanding piecewise functions helps in analyzing and predicting costs or outcomes in such scenarios.
The monthly cost function E ( x ) for electricity usage can be expressed as a piecewise function: 300 \end{cases}"> E ( x ) = { 4 + 0.10 x 0.26 x − 44 if 0 ≤ x ≤ 300 if x > 300 . This accounts for the base rate and different usage rates for the first 300 kWh and any amount exceeding that.
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