Westside Energy charges its electric customers a base rate of $4.00 per month, plus 10¢ per kilowatt-hour (kWh) for the first 300 kWh used and 2¢ per kWh for all usage over 300 kWh. Suppose a customer uses [tex]$\times kWh$[/tex] of electricity in one month.

(a) Express the monthly cost [tex]$E$[/tex] as a piecewise defined function of [tex]$x$[/tex]. (Assume [tex]$E$[/tex] is measured in dollars.)

[tex]$E(x)=\left\{\begin{array}{ll} //, & \text { if } 0 \leq x \leq 300 \\ //, & \text { if } 300\ \textless \ x \end{array}\right.$[/tex]

(b) Graph the function [tex]$E$[/tex] for [tex]$0 \leq x \leq 600$[/tex].

Define the cost function for 0 ≤ x ≤ 300 as E ( x ) = 4 + 0.10 x .
Define the cost function for 300"> x > 300 as E ( x ) = 28 + 0.02 x .
Express the monthly cost E as a piecewise defined function of x : 300 \end{cases}"> E ( x ) = { 4 + 0.10 x , ​ if 0 ≤ x ≤ 300 28 + 0.02 x , ​ if x > 300 ​ .
Graph the piecewise function with a line segment from ( 0 , 4 ) to ( 300 , 34 ) and a line segment from ( 300 , 34 ) to ( 600 , 40 ) .

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 a month. The base rate is $4.00. The cost is 10¢ per kWh for the first 300 kWh and 2¢ per kWh for usage over 300 kWh.

Expressing the Cost Function For 0 ≤ x ≤ 300 , the cost is the base rate plus 10¢ (or 0.10 ) p er kWh . T h ere f ore , t h ecos t f u n c t i o ni s : E ( x ) = 4 + 0.10 x F or x > 300$, the cost is the base rate plus 10¢ per kWh for the first 300 kWh and 2¢ (or $0.02) per kWh for the usage over 300 kWh. The cost for the first 300 kWh is $0.10


imes 300 = 30$. The cost for the usage over 300 kWh is 0.02 ( x − 300 ) . Therefore, the cost function is: E ( x ) = 4 + 30 + 0.02 ( x − 300 ) = 4 + 30 + 0.02 x − 6 = 28 + 0.02 x

Defining the Piecewise Function So, the piecewise defined function is: 300 \end{cases}"> E ( x ) = { 4 + 0.10 x , ​ if 0 ≤ x ≤ 300 28 + 0.02 x , ​ if x > 300 ​

Graphing the Function To graph the function for 0 ≤ x ≤ 600 , we need to plot the two linear equations for the specified ranges of x .
For 0 ≤ x ≤ 300 , the equation is E ( x ) = 4 + 0.10 x . When x = 0 , E ( 0 ) = 4 . When x = 300 , E ( 300 ) = 4 + 0.10 ( 300 ) = 4 + 30 = 34 . So, we have a line segment from ( 0 , 4 ) to ( 300 , 34 ) .
For 300 < x ≤ 600 , the equation is E ( x ) = 28 + 0.02 x . When x = 300 , E ( 300 ) = 28 + 0.02 ( 300 ) = 28 + 6 = 34 . When x = 600 , E ( 600 ) = 28 + 0.02 ( 600 ) = 28 + 12 = 40 . So, we have a line segment from ( 300 , 34 ) to ( 600 , 40 ) .


Examples
Understanding piecewise functions like this helps in real-world scenarios such as calculating costs based on usage tiers. For example, cell phone plans often have different rates for data usage, where the cost per gigabyte changes after a certain threshold. Similarly, utilities like water and gas use tiered pricing to encourage conservation. By modeling these scenarios with piecewise functions, we can predict and manage our expenses more effectively.

Answered by GinnyAnswer | 2025-07-08

The monthly cost function E ( x ) is defined in two parts based on usage: E ( x ) = 4 + 0.10 x for 0 ≤ x ≤ 300 and E ( x ) = 28 + 0.02 x for 300"> x > 300 . The graph consists of two linear segments reflecting these cost calculations. This piecewise function effectively models the varying costs of electricity usage from Westside Energy.
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Answered by Anonymous | 2025-07-17