A 250-watt television and a 400-watt television are both used for two hours. How much more energy does the 400-watt television require to operate for this time than the 250-watt television?
Alright, so this question, we’ve got a 250-watt television and a 400-watt television and we’ve been told that they’re both used for two hours. What we need to do is to find out how much more energy the 400-watt television uses compared to the 250-watt television when they’re left on for the same period of time.
Now, the 250 watts and 400 watts tell us the power of the television and we’ve been told the amount of time for which the television is left on. As well as this, we need to find out the amount of energy used by each television and more specifically how much more energy is used by the 400-watt television.
So we need to find a relationship that links together the powers of the television, the time for which they’re left on, and the energy used by the televisions. The relationship that we’re looking for is that the power 𝑃 is equal to the energy 𝐸 divided by the time 𝑡. In other words, power is the rate of change of energy or how much energy is for example used by an appliance for every unit of time.
Now, let’s quickly discuss the standard units of each one of these quantities. The standard unit of power is watts, for energy, it’s joules, and for time, it’s seconds. Now, since we’re trying to find out what the energy usage of each one of these televisions is and ideally we want to give the answer in the standard unit of joules, then we need to work with power in watts and time in seconds.
Now, as for the power in watts, we’ve already got that. We’ve been told that the powers of the televisions are 250 watts and 400 watts, respectively. However, the amount of time for which the televisions are used is given in hours. We’re told that the TVs are used for two hours. So what we need to do is to convert this time of two hours into seconds.
Let’s start by saying that our time 𝑡 is equal to two hours. And we can recall that one hour has 60 minutes in it and one minute has 60 seconds in it. Therefore, we can use this information to work out how many seconds there are in one hour. So in one hour, there are 60 minutes and every minute has 60 seconds in it. So there are 60 times 60 seconds in one hour. And 60 times 60 is 3600. So every hour has 3600 seconds in it. But the amount of time for which we use the television is two hours, not one hour. So to work out the number of seconds in two hours, we say that this is simply equal to two times 3600 seconds and that ends up being 7200 seconds.
So now, we’ve got the amount of time for which the TVs are used in seconds. Let’s write that down here and move on. Now, if we want to work out how much more energy the 400-watt television uses compared to the 250-watt television, then first, we need to work out how much energy each one of the televisions uses. In order to work out how much energy is used when we’re given the power and the time, we need to rearrange this equation here.
What we need to do is to multiply both sides of the equation by the time 𝑡. Well, what that leaves us with is that the power multiplied by the time 𝑡 is equal to the energy 𝐸. Now, we can use this equation to work out the energy used by the 250-watt TV and the 400-watt TV. Let’s say that 𝐸 sub 250, the energy used by the 250-watt TV, is equal to the power of the 250-watt TV which is 250 watts multiplied by 7200 seconds which is the time for which the TV is used.
Now, similarly, for the 400-watt TV, the energy used which is 𝐸 sub 400 is equal to the power which is 400 watts multiplied by the time for which it’s used which is again 7200 seconds. Now, we can evaluate what 𝐸 sub 250 and 𝐸 sub 400 are. We find that 𝐸 sub 250 is 1.8 times 10 to the power of six joules and 𝐸 sub 400 is 2.88 times 10 to the power of six joules.
But this is not our final answer. We need to find how much more energy the 400-watt television uses compared to the 250-watt television. In other words, we need to find 𝐸 sub 400 minus 𝐸 sub 250. This quantity is equal to the difference in the two energies. And therefore, that’s the amount of extra energy that the 400-watt television uses compared to how much the 250-watt television uses.
So we find that 𝐸 sub 400 minus 𝐸 sub 250 is equal to 2.88 times 10 to the power of six minus 1.8 times 10 to the power of six, which simplifies to 1.08 times 10 to the power of six joules.
And this is our final answer. The 400-watt television uses 1.08 times 10 to the power of six joules of energy more than the 250-watt television.