3-3 Cost of Energy

Review: Units of Power and Energy

Power is defined as how much energy is used per unit of time:

If we rearrange the equation to solve for energy, we get this:

When we're looking at a word problem, we need to make a decision about what set of units to use: base SI (metric) units, or "convenient" units for cost.

Here's when to use which set of units.

Use base units when...

• it's a small device, like a single light bulb or small appliance

• it's operating for a short period of time, on the order of seconds

• there's going to be a small amount of power involved, measured in watts

Use convenient units when...

• it's a large device or a whole house

• it's operating for a long time, on the order of hours or days

• there's a big amount of power involved, measured in kilowatts (or bigger)

How much does electricity cost?

The short answer is: it depends. (But sometimes a short answer isn't quite enough, is it?) Either way, though, people pay for their energy by the kilowatt-hour, not the joule. That would just be silly.

A lot of people in Ontario buy their energy using time of use (TOU). That means that, depending on when the energy is used, it will cost a different rate. As you can see, the price also depends on what time of year it is.

There are three different prices for electricity:

• on-peak is the highest price... this happens when the demand for electricity is the highest

• off-peak is the lowest price... this happens when demand is lowest

• mid-peak is in the middle for both, as the name suggests

All of this is driven by the idea of supply and demand: when the demand for something is the highest, and/or there is a smaller supply of something, the price will be higher. This is true for a lot of things, and in Ontario a lot of people buy their electricity this way.

For example, dishwashers will often have a "Delay Start" button. Here's a button that allows you to delay the start of the dishwasher cycle by either two or four hours:

If you're washing a load of dishes after dinner, does it really matter if they're being washed at 6:30 pm, which might be either mid- or on-peak, or at 11:00 pm, when it's definitely off-peak and therefore cheaper? Probably not, so go ahead and save some money. Pennies turn into dollars pretty quickly.

Calculating cost of energy

Example: A microwave oven has a power of 1.3 kW, and over a month it is used for a total of 6.5 h.

(a.) Find the toal amount of energy used.

(b.) If energy costs $0.15 for each kW·h, find how much it cost to run the microwave oven.

- - - - - - - - - -

(a.) Here we use the formula P = E/t, but we rearrange it and use the E = Pt version. Also, we look at the units and decide that it's better to use the "convenient" units of kW·h, kW and h.

P = 1.3 kW

t = 6.5 h

E = ?

E = Pt = (1.3 kW)(6.5 h) = 8.5 kW·h

(b.) We can think this one through without using a formula, really... if each kW·h costs $0.15, and we buy 8.5 of them...

total cost = ($0.15)(8.5) = $1.28 (rounded to the nearest cent)

- - - - - - - - - -

As of December 2022, here are the prices in Ontario for time-of-use (TOU) electricity:

Note that these prices are given in cents per kilowatt-hour, not dollars. Be aware of this when doing these calculations.

Example: In a week, a household uses 78 kW·h of off-peak energy, 35 kW·h of mid-peak energy, and 122 kW·h of on-peak energy. Calculate the total cost of the electricity, given the prices in the chart above. Express the answer in dollars, rounding to the nearest cent.

- - - - - - - - - -

A good approach is to do the small calculations in pieces, then put the pieces together to get a final answer.

Off-peak cost: (7.4¢)(78) = 577¢ = $5.77

Mid-peak cost: (10.2¢)(35) = 357¢ = $3.57

On-peak cost: (15.1¢)(122) = 1842¢ = $18.42

Total cost: $5.77 + $3.57 + $18.42 = $27.76

Practice

The basics

1. A clothes dryer has a power rating of 2.8 kW. If it is on for a total of 39 h, find the energy used by the dryer.

2. If the cost of electricity is 7.4¢ per kilowatt-hour, determine the cost to use the dryer in the previous question. Answer in dollars, rounded to the nearest cent.

3. When is electricity the cheapest: at 9:00 pm in March, noon in August, or 4:00 pm in December? When is it the most expensive out of those three choices?

Extensions

1. Are there any other major appliances in your home that have a time-delay feature? (Hint: look in the laundry area.)

2. Look at when the on-peak pricing occurs in the summer. What do you think is the major reason for this?