Question

A water company

charges residential customers a quarterly flat fee of
$25 for up to 5000 gallons of water. The company
charges $10 per thousand for the first 10,000
gallons of water after the minimum allowance. The
next 10,000 gallons is $5 per thousand. All other
consumption over 25,000 gallons is $2.50 per
thousand.
a. Write a piecewise function w(x) that can be
used to determine a customer’s quarterly bill for
using x gallons of water.

Answer 1

The piecewise function w(x) to determine a customer's quarterly water bill is provided: w(x) = 25 + (10/1000) * (x - 5,000) if 0 ≤ x ≤ 5,000, w(x) = 25 + (10/1000) * 10,000 + (5/1000) * (x - 15,000) if 5,000 < x ≤ 15,000, w(x) = 25 + (10/1000) * 10,000 + (5/1000) * 10,000 + (2.50/1000) * (x - 25,000) if x > 25,000.

Step-by-step explanation:

To write a piecewise function w(x) that can be used to determine a customer's quarterly bill for using x gallons of water, we need to define the different cases based on the amount of water used.

For the first 5,000 gallons, the flat fee of $25 applies.

For the next 10,000 gallons after the minimum allowance, the cost is $10 per thousand gallons. So, the cost for this range is $(10/1000) * x.

For the next 10,000 gallons, the cost is $5 per thousand gallons. So, the cost for this range is $(5/1000) * (x - 10,000).

For all consumption over 25,000 gallons, the cost is $2.50 per thousand gallons. So, the cost for this range is $(2.50/1000) * (x - 25,000).

Putting it all together, the piecewise function w(x) can be written as:

w(x) = 25 + (10/1000) * (x - 5,000) if 0 ≤ x ≤ 5,000

w(x) = 25 + (10/1000) * 10,000 + (5/1000) * (x - 15,000) if 5,000 < x ≤ 15,000

w(x) = 25 + (10/1000) * 10,000 + (5/1000) * 10,000 + (2.50/1000) * (x - 25,000) if x > 25,000