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How do you write a piecewise function from a word problem

It costs 7.57 plus 8.77 per kilowatt hour for the first 400 and 6.574 per kilowatt usage over 400 in a month.

What is the cost of 300 in a month? 2638.57 how much for 700 in one month? 5487.77 What I need is a piecewise function for this x=Kwhr used a month and c= cost of usage

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3 Answers

The second piece of your function is going to be for when the kilowatts used is over 400, or when x > 400; since the first piece of your function was for x <= 400.

As the problem stated, it costs 6.574 per kilowatt hour usage after 400. So if x is your total number of kilowatts used, then (x-400) will be the total number of kilowatts used after 400. So multiply 6.574 by (x-400):

6.574(x-400)=

6.574x-2629.6

This is only the cost for the hours used after 400. But we are looking for the total cost, so we will need to add the cost of the kilowatt hours used before or equal to 400 to the cost of the hours used after 400. We already found the cost of the kilowatt hours used before or equal to 400, this is just the first piece of your function that George P. already helped you solve. So we take:

8.77*400+7.57 = 3515.57

and add this to our "after 400" cost. So then we have:

3515.57+6.574x-2629.6 =

6.574x+885.97. This is the total cost when x > 400.

So your piecewise function should look like:

C(x) = 8.77x+7.57 if x <=400

           6.574+885.97 if x > 400

You are right, you need a piecewise function, which you can separate based on the the segments of the domain that are given in words...

First, there is the the segment from 0-400 kwh, then the segment from 400 - infinity (what an electric bill!)

 

Here is how to write it:

 

                    7.57 + 8.77x  ,     0<= x < 400 kilowatt hours

f(x) =            (5.57 + 877*400) + 6.574(x - 400),     x > 400 kilowatt hours

 

Imagine a great big "curly brace" to the left of the equal sign, that wraps onto the two different functions.

The trick is to realize that the first 400 kwh translate into a constant that is added into rate*time of the amount over.  You also have to correct x in the second equation, since you don't want to charge twice  for the first 400 hours

                    

 

First, think about what you did to figure out the first answer when the number of kilowatt hours was 300.

C, the cost of usage, is what you found, when you used 300 as your kilowatt hours.

So, you multiplied 8.77 by 300 and added 7.57

So,  C  =             8.77 x         +             7.57     , but this is only if the kWh or x is not more than 400

Or, the first part of your function is C = 8.77x + 7.57  if  x <= 400.

 

Now, you need to find the part of the piecewise function when x is greater than 400:

So, for this part of the function, think about how you found $5487.77 for 700 kWh.

Think about what you did, and in place of the 700, replace it with x.

Can you figure this part out?

 

Comments

Well, were you able to see how to get the other part of your function?


First, how much will the cost be with 400 kWh?

Then, how do you find the additional cost above 400 kWh?

 

To find the cost at 400 kWh:

C = 8.77 (400)  + 7.57  =  3515.57

 

If you were to use 700 kWh, you would have an additional 300 kWh,

(700 - 400)

You would multiply this by the new rate of 6.574

 

So, for 700 kWh, the total amount would be C = (700-400)6.574 + 3515.57

So, in general, for an x amount of kWh above 400 kwh, you

would have C = (x-400)6.574 + 3515.57.

This would be the part of the function for x > 400.

 

So, here is your piecewise function:

                                 8.77x  +  7.57               if x <= 400

C    =

                                6.574(x - 400)               if  x  >  400