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).

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 being added to 17. Where x is the number of days after the share is launched. And we want to determine the period of p of x. For our answer choices, a is 113 days, b is 226 days, c is 339 days, and d is 452 days. Now, what are we trying to figure out here? We want the period of p of x. And notice that p of x is a trigonometric function. Recall that the general form for a trigonometric function is a multiplied by the function. In this case, the sign of b x minus c plus d, where we can derive different things from our function based on the values of a, b, c, and d. But right now, we're only interested in the period and the period for our function. The period Would be equal to 2 pi divided by b. So, if we can figure out the value of b from our function, then we should be able to figure out the period for the number of for x being the number of days after the share is launched. Now, just looking at it, the value of x the real value of x might not seem readily available. But with a little working out, we can figure it out. So notice that the sign of our angle. Is the expression pi divided by 1 113 multiplied by x minus 73. If we distribute pi divided by 113, then we'll get pi multiplied by x divided by 113 minus 73 multiplied by pi divided by 113. In this case, notice that the coefficient of x is pi divided by 113. Therefore, this tells us that b equals pi divided by 113. And now that we know the value of b, that means the period Would be equal to 2 pi divided by B. So that's 2 pi divided by pi divided by 113. We can rewrite this as 2pi multiplied by 113 divided by pi. And now, both terms have a common factor of pi that we can cancel. Therefore, we're left with 2 multiplied by 113, which equals 226. Therefore, the period for our function p of x is 226 days. And when we go back to our list of choices, that would have been answer choice b. 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. It has a period of 365 days.

Key Concepts

Trigonometric Functions

Periodicity

Phase Shift

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