Temperatures in Fairbanks, Alaska The graph i...

Question

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

a. On what day is the temperature increasing the fastest?

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Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. In a study of the city's traffic flow, the volume of cars on a major highway is modeled by the equation. V equals 730 divided by pi multiplied by sign of pi divided by 365 multiplied by parentheses D minus 80 plus 150, where V is the volume of cars and D is the day of the year. On what day is the volume of traffic increasing the fastest? Awesome. So that's our angle. Our angle, our final answer we're ultimately trying to solve for is we're trying to figure out on what exact day is the volume of a traffic increasing the fastest. So with that said, now that we know what we're ultimately trying to solve for, let's read off our multiple choice sensors to see what our final answer is, knowing that they all say it's some day. So it's on a specific day, so it's gonna be blank. So A is 365, B is 245, C is 160, and D is 80. Awesome. So, first off, in order to solve this particular problem, we need to differentiate the traffic volume function V with respect to D. So we need to take Vas V indicates the first derivative, so we need to take V is equal to D by DD because we're taking the derivative with respect to D. Of 730 divided by pi multiplied by of parentheses, divided by 365, multiplied by parentheses D minus 80. Plus 150, fantastic, and the plus 150 is outside of the parentheses. So, when we start to perform our derivative, we will find that V is going to be equal to 730 divided by pi multiplied by cosine of. I divided by 365 multiplied by D minus 80. Multiplied by D by DD of pi divided by 365, and when you take pi divided by 365, we need to multiply it by parentheses, D minus 80, so multiplied by D minus. 80 And then plus 0. So we're still trying to simplify and perform our derivative and solve for V. So V is gonna be equal to 730. Divided by pi, multiplied by cosine of pi divided by 365, multiplied by parentheses D minus 80. Multiplied by divided by 365. And then finally simplifying one last time, you will find that V is going to be equal to 2 multiplied by cosine of pi divided by 365, multiplied by D. Minus 80 in parenthesis, so. D minus 80 in parenthesis. Fantastic. So we need to recall and note that the derivative will reach its maximum when the cosine function is equal to 1. So we need to solve for D when the cosine function is equal to 1, so we need to take cosine of pi divided by 365, so you need to take pi divided by 365. Multiplied by D minus 80 is equal to 1. So, when we start to solve, we will find that pi divided by 365. So when you take pi divided by 365, and we need to multiply it by parentheses D minus 80, it's going to be equal to inverse cosine of 1. Awesome. So we now need to note or make note of the general solution for so inverse cosine of 1 is to high end, so let's make a note of that in blue. So let us note that Inverse cosine of 1. I 2 N. Fantastic. So with that in mind, let us note that inverse cosine of 1 is going to be equal to curly brackets of 0.24. Inside curly brackets. So therefore we can go ahead and note that pi divided by 365, multiplied by parentheses D minus 80 is going to be equal to 0.24. So that will mean that D minus 80 is equal to 0,730, 1460. So that will mean that isolating and solving for D, D will be equal to 80. So once again, we're when we're rearranging the isolates sour D, D will be equal to 80,810. 1540. So that means that since D represents the day of the year, so let's make a note that 1 is less than or equal to D and D is less than or equal to 365. The only valid solution will have to be D is equal to 80 days or day 80. So therefore, the volume of traffic is increasing the fastest on day 80 of the year. So, Therefore, D is equal to 80, so our final answer has to be day 80. So that will mean when we look at our multiple choice answers, the correct answer, without a doubt has to be multiple choice answer letter D, which once again is day 80. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Derivative and Rate of Change

Sine Function and its Properties

Chain Rule for Differentiation

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