{Use of Tech} Power and energy The total energy in megawatt-...

Question

{Use of Tech} Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

Accepted Answer

Welcome back, everyone. In this problem, a city's water pumping station operates based on the energy consumption model. E of T equals 500T plus 1800 divided by pi multiplied by the cosine of pi T divided by 6, where E of T is the total energy in kilowatt hour and T greater than or equal to 0 is measured in hours from midnight. At what time of day does the energy consumption rate first reach its minimum, and what is the minimum rate of energy consumption? A says it reaches its first minimum at 9 a.m. and the minimum rate of energy consumption is 200 kilowatts. B says it's 3 a.m. and 200 kilowatts. C 9 a.m. and 800 kilowatts, and D 3 a.m. and 800 kilowatts. Now, in both of the questions that we would like to answer, we're trying to understand the point at which the energy consumption rate, sorry, the energy consumption rate reaches its minimum, OK? And once we know where that minimum rate is, we can find that the minimum rate of energy consumption. Now to find a minimum, let's first start by differentiating E of T with respect to T to find the rate of energy consumption in kilowatts, OK? So no Given that we already know he of tea. Is equal to 500T. Plus 1800 divided by pi multiplied by the cosine of sorry, T divided by 6, OK. Then to differentiate this term. We can differentiate 500T using the poor rule, and then 1800 divided by pi cost pi T divided by 6 using the chain rule, OK? So let's go ahead and do that. Now, let's, let's start with the harder one, which is our second term, OK? Because no, for E of T, OK, then to find a derivative with respect to T. Of 1800 divided by pi plus by divided by 6, OK. Then by the chain rule we know that what it means. Is that first, uh, we are going to Multiply, well, we're still gonna have our constant. Let me, let me write it over here. We're still gonna have our constant 1800 divided by pi. Next, we know the derivative of a cosine is negative sign, so that's negative sine IT divided by 6, and then we're going to multiply this by the derivative of our inner function pit divided by 6. And here when we differentiate that by the poor rule, it would leave us with pi divided by 6, OK. And no notice here that we can cancel supply. And we can say 6 goes into 1800, 300 times. So now, the derivative of this expression is going to be negative 300 multiplied by the sign of IT, OK. Divided by 6. And know that we have that, that means then the derivative of E of T, OK, is going to be equal to the derivative of 500 T, which is 500 plus this expression, which is going to be negative 300 sign of PT divided by 6. OK. So this is the derivative of E of T. Now here, notice that we have the sine function in our expression and the sine function oscillates between -1 and 1. Now, since E of T decreases as the sine function increases, that means the minimum value of A of T, OK, so let's write that, the minimum value. Of EFT or sorry, the derivative of EFT rather. It occurs. When the sign of. IT divided by 6 is at its maximum value, which is equal to 1, OK? Now, when does the sign function reach one? If we figure that out, it can help us to figure out what time of day the energy consumption rate first reaches its minimum, OK? No. The same function Um, let's add that here. So we're the same function. Reaches its reaches one, OK. And function reaches 1 at the point where our argument is equal to pi divided by 2 plus a multiple of 2 pi. In other words, 2 and pi, where N is an integer. OK, so that's gonna be the case. Now at what multiple will it first reach its minimum? Well, we know that that will be when N equals 0, OK? And no. If that's the case, then when N equals 0. OK. Then pi T divided by 6 is going to be equal to pi divided by 2. If we cancel pi from both sides and multiply both sides by 6, then that means T is going to be equal to 3, OK? Now what does this 3 represent? Well we're talking about time in hours. Therefore, T equals 3 hours. Remember in our problem statement. OK. We said that T was measured in hours from midnight. So that means T here at the minimum is gonna be 3 hours from midnight, OK. So that tells us the first time, I think that might be a lot to write, but the energy consumption rate, energy consumption rate. Reaches its first minimum. 3 hours from midnight, OK, which is going to be 3 a.m. OK. Because we got T to be equal to 3, and remember we're measuring T starting from midnight. Know that we've found at the time at which it reaches the first minimum, what is the minimum rate of energy consumption? Well, remember earlier we said that the minimum value of the rate occurs when the sign of pit divided by 6 is equal to 1. So now, if we plug that value into our rate, OK, that means our rate is going to be equal to 500. -300 multiplied by 1 we can replace the sign of piety divided by 6 by 1. And no, that's gonna be 500 minus 300, which equals 200 kilowatts. Therefore, the time at which The energy consumption rate reaches its first minimum is 3 a.m. and the minimum rate of energy consumption is 200 kilowatts. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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