### Video Transcript

The London Array is a wind farm located off the coast of Kent in the Thames Estuary. Its installed capacity is 630 megawatts. What is the total energy output of the London Array over the course of a year if it runs at 50 percent of its installed capacity throughout the year? Use a value of 365 for the number of days in a year. Give your answer in gigawatt hours to two significant figures. A) 2800 gigawatt hours, B) 3600 gigawatt hours, C) 630 gigawatt hours, D) 50 gigawatt hours, or E) 1200 gigawatt hours.

A wind farm is a collection of wind turbines. And wind turbines convert kinetic energy from the air into electrical energy. This question talks about the installed capacity of a wind farm. The installed capacity is the maximum amount of energy that the wind farm is capable of producing. In other words, it’s the amount of electrical energy that the wind farm would produce if there were enough wind to keep all of the wind turbines constantly turning at their maximum speed. In reality, however, we won’t have ideal conditions all the time. So, we find that wind farms actually produce less than their installed capacity.

In this question, we’re looking at a wind farm with an installed capacity of 630 megawatts. Let’s call this 𝑃 sub cap. We’re also told that this wind farm operates at 50 percent of its installed capacity. And we’re asked to find out how much energy is generated by the wind farm in total in a year or 365 days, to be exact. Let’s call this period of time 𝑡.

Let’s start by working out the actual average power output of this wind farm. If the installed capacity is 630 megawatts and the wind farm produces 50 percent of that. Then, the actual power output, 𝑃, is 50 percent of 630 megawatts or 0.5 times 630 megawatts, which is 315 megawatts. Okay, so now we have the power output of the wind farm and the time for which it’s outputting that power. And we want to find the total energy output, which we’ll call 𝐸.

Recall that energy, power, and time are related by the equation 𝐸 equals 𝑃𝑡, energy equals power times time. Before we substitute our values for 𝑃 and 𝑡 into this equation, we need to make sure that we’re using the correct units. All of our answer options are expressed in gigawatt hours. In order to make sure that our answer is expressed in gigawatt hours, we need to make sure that our value of 𝑃 is expressed in gigawatts and our value of 𝑡 is expressed in hours. This way, when we multiply power and time together to get energy, we’ll be multiplying a quantity in gigawatts by a quantity in hours. So, the result will be in gigawatt hours.

Let’s start by expressing 𝑡, 365 days, in hours. There are 24 hours in a day, so 365 days contain 365 times 24 hours, which is 8760 hours. Next, let’s convert our power, 315 megawatts, into gigawatts. Recall that adding the mega- prefix to any unit increases its size by a million times. So, one megawatt is a million watts.

The prefix giga- increases the size of a unit by a billion times or 10 to the power of nine. So, one gigawatt is equivalent to one billion watts. So, we can see that one gigawatt is 1000 times bigger than a megawatt. This means that 315 megawatts is 1000 times smaller than 315 gigawatts. Or, in other words, to express this quantity in gigawatts, we have to divide it by 1000, which gives us 0.315 gigawatts.

Now that our quantities, 𝑃 and 𝑡, are expressed in terms of the correct units, we can substitute them into our formula, which gives us 0.315 times 8760, which is equal to 2759.4 gigawatt hours. The question asks us to express our answer to two significant figures. The first two significant figures in our answer are two and seven. And the third figure tells us whether or not we need to round up. We should round up if this figure is five or more.

Since it’s a five, we round up the previous figure to an eight. So, our answer to two significant figures is 2800 gigawatt hours. This means the correct answer is option A. If a wind farm with an installed capacity of 630 megawatts runs at 50 percent of its installed capacity for a year, then the total energy output is 2800 gigawatt hours .