Daylight function for 40 °N Verify that ...

Question

Daylight function for 40 °N Verify that the function $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

$D of 81 equals 12$ and $D of 264 almost equals 12$ (corresponding to the equinoxes).

Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at $t equals 172$ and $t equals 355$ , respectively (corresponding to the solstices).

Accepted Answer

Welcome back everyone. In this problem, the price of a share is given by the equation p of x equals 3.7 multiplied by the sign of the product of pi divided by 113 and x minus 73 plus 17. Where x is the number of days after the share is launched. And we want to determine the minimum and maximum value of p of x along with the respective values of x. For our answer choices, we have the minimum and maximum values of x and p of x which are all some form of a combination between the numbers 16.5, 129.5, 13.3, and 20.7. Now let's think a little bit about this. So we have the price of the share and we want the minimum and maximum values of p of x along with the respective values of x. Let's think about our function. P of x is a trigonometric function. As a matter of fact, it's a sine function. What do we know about the sine function? Well, recall that for the sine function, first thinking about the maximum value, recall that the sine function has a maximum value of 1. It can't be greater than 1. So that means at the maximum value or sine function would be equal to 1. So we can use that idea to figure out what our value of p of x would be. So at the maximum value. At the maximum value. The sine of theta would be equal to 1, which then tells us that p of x would be equal to 3.7 multiplied by our sine function. Remember, it doesn't matter what's here. We know that the value is going to be 1. So that's 3.7 multiplied by 1 plus 17. That would be equal to 3.7 +17 which is 20.7. So that's the maximum value of p of x. Now, how about X? Well, we just said it. We said the maximum value of the sine function equals 1. So therefore, at the maximum value, it also means That or sine function came or sine function. Which is the sign of I divided by I should write that properly. Pi divided by 113 multiplied by x minus 73. Would be equal to 1 at the maximum value. So now we can see if we can transpose and use that to help us to figure out the value of x. Now, if the sine function is 1, can you recall what what value of sine gives us 1? Well, it would be half of pi. So since the sine of a half of I should I should write that because that's something we know from our unit circle. So since the sine of a half of pi equals 1, what does that tell us? It tells us then that our angle, our value inside the sine function would be equal to a half of pi. So that implies then that Pi divided by 113 multiplied by X minus 73 equals a half of Pi. Now, if we do some, transposition here, I can divide both sides by I yeah. I can divide both sides by Pi divided by 113. So x minus 73 put that back in red. X minus 73 would be equal to pi divided by 2 divided by pi divided by 113, which I can rewrite as being multiplied by 113 divided by pi, where the values of pi cancel. So that means x minus 73 equals 113 divided by 2. 113 divided by 2 is 56.5. So therefore x would be equal to 56.5 plus 73, which equals 129.5. So therefore, that tells us that at the maximum value, p of x equals 20.7 and x would be equal to 129.5. So we know our values at the maximum. How about at the minimum? Well, if we think about it again, what else do we know about the sine function? Recall that the sine function has to be greater than or equal to negative one. It can't be less than negative one. So, therefore, that tells us that the minimum value of the sine function, the sine of theta equals negative one. Now, what do we know about the sine of theta and negative one? Well, if our value is negative one, we can solve for p of x because that now tells us let me put that in a different color. Let me put that in blue. That now tells us that p of x would be equal to 3.7 multiplied by our sign function at the minimum value which is negative one plus 17. That equals 17 minus 3.7 which equals 13.3. So at the minimum value, p of x equals 13.3. How about our value of x? Well, we can use the same same idea as before. We can substitute for our value of the sine of theta. So. At the minimum value, that also tells us that our sine function, the sine of pi divided by 113 multiplied by X minus 73. All of that would be equal to negative one. We can recall that the sign of negative a half of pi equals negative one. So that can help us to solve because now it tells us that our angle. Would be equal to pi divided by 1 113 multiplied by X minus 73, that would be equal to negative a half of pi. Again, we can let me draw a little border here to show that I'm coming over on the right because now this tells us that x minus 73 would be equal to negative pi divided by 2 multiplied by 113 divided by pi. In other words, we divided by pi divided by 113 like we did previously. Pi cancels pi. So that means x minus 73 equals negative 113 divided by 2, which is negative 56.5. And now that tells us that x would be equal to 73 minus 56.5, which equals 16.5. Therefore, at the minimum value, p of x equals 13.3 and x equals 16.5. Now when we look back in our answer choices, notice, as we said, at the maximum payoff x equals 20.7 and x equals 129.5. And at the minimum, p of x equals 13.3 and x equals 16.5. That means our answer would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Trigonometric Functions

Periodicity

Maxima and Minima

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