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).

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

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 and all of that is added to 17. And x is the number of days after the share is launched and we want to determine the first two positive values of x when p of x equals 17. For our answer choices, a is x equals 129.5186, b is x equals 16.5 and 73. C is x equals 16.5 and 129.5. And the d is x equals 73186. Now, what do we know here? Well, we know what we want to find the first two positive values of x when p of x equals 17. We don't know what x is, but we know that p of x equals 17 and we have a function for p of x. So if we can substitute that value of p of x, we should be able to solve for x in our function. So let's go ahead and do that. So substituting p of x equals 17. Then that means our function, which is 3.7 oops. 3.7 multiplied by the sign of the product of pi divided by 113 and x minus 73 plus 17 equals p of x, which is 17. Now, notice both sides have 17, so we can subtract 17 from both sides which means that 3.7 multiplied by the sine of our angle equals 0. Now, notice that 3.7 is multiplying the sign of the angle, so we can divide both sides by 3.7 as well. And 0 divided by 3.7 equals 0. So now, we have the sine of 1 pi divided by 113. This this is really a lot to say. Multiply it by x minus 73, all of that equals 0. Now, where do we go from here? Well, let's think about it. We are saying that the sine of some angle, let's call it theta equals 0. If we think about the value of sine on our unit circle, what we're essentially asking ourselves is what angle will make the sine of theta be equal to 0. Now, recall that for the unit circle, the x and y coordinates are in the form the cosine of theta and the sine of theta where the cosine of theta is the x coordinate and the sine of theta is the y coordinate. So, if the sine of theta equals 0, we're looking on our unit circle where the y value is 0, which it would be here where our angle is 0 radians. That would be our first value. And then, it would also be at the point where our angle is pi radians because it would be right here on our unit circle. So you know what that tells us. Therefore, that means that if the sine of our angle equals 0, then our angle which is pi divided by 113 multiplied by x minus 73 equals 0 and it equals pi. The first two positive values for our sine function when the sine of the angle equals 0. So, now what that means is that we know all we need to do to solve for x is to equate both to those values. So, therefore, that tells us that our product pi divided by 113 multiplied by x minus 73 equals 0 and pi divided by 113 multiplied by x minus 73 equals pi. So now, all we need to do is to solve for x in both. So, let's start with our first one. Since we are multiplying x minus 73 by pi divided by 113 and pi divided by 113 is a constant, we can divide both sides by pi divided by 113. So when we do that, divided by pi divided by 113 to cancel out. Then, on our left hand side, we'll be left with x minus 73 and 0 divided by pi divided by 113 equals 0. If x minus 73 equals 0, that means x must be equal to 73. So that's our first value of x. Let's solve in the second case. Now, if pi divided by 113 multiplied by x minus 73 equals Pi, since both sides are being multiplied by Pi, we can divide both sides by Pi. Okay? And when we divide both sides by Pi, pi cancels pi on the left hand side and pi also cancels pi on the right hand side, which leaves us with 1 divided by 113 multiplied by x minus 73 on the left hand side and one on the right hand side. Now we can multiply both sides by 113. Okay. Because remember, we want x by itself. So we're doing everything we can to make that possible. And if we multiply both sides by 113, they cancel on the left hand side, which leaves us with x minus 73. And on the right hand side, we'll have 113. If x minus 73 equals 113, then x equals 113 plus 73 which equals 186. Therefore, for our function p, x equals 73 and x equals 186 when p equals 17. When we look back in our answer choices, that would have been answer choice d. Thanks a lot for watching everyone. I hope this video helped.

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).

Key Concepts

Trigonometric Functions

Periodic Functions

Function Evaluation

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