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.

b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

<IMAGE>

Accepted Answer

Welcome back, everyone. In a study of a 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 the sign of pi divided by 365 multiplied by D minus 80 plus 150, where V is the volume of cars and D is the day of the year. About how many cars per day is the traffic volume increasing when it is increasing at its fastest? A says 2 cars per day, B 3 cars per day, C 4 cars per day, and D, 5 cars per day. Now to find how many cars per day, the traffic volume is increasing when it is increasing at its fastest, we can differentiate V with respect to D to figure out how our traffic volume is increasing, and then solve for the point where it is the maximum value. That is when it is increasing at its fastest. So let's go ahead and do that. First, let's differentiate V. With respect to the eh. Now that means then we're going to be finding the derivative of this expression, and when we do that, we should get the derivative of V to be equal to notice here that this is our composite function, the first term in our equation for V. So by differentiating it using the poor rule. It's going to be 730 divided by pi multiplied by the cosine of pi divided by 365 multiplied by D minus 80 multiplied by the derivative with respect to D of our inner function that is pi divided by 365 multiplied by D minus 80, OK, plus 0 because the derivative of 150, a constant is 0. Now if we go ahead here and differentiate our inner function. Then we should find that the derivative of V equals 730 divided by pi. Multiplied by the cosign of. I divided by 365, multiplied by D minus 80, OK. Multiplied by pi divided by 365. And now if we do some simplifying here, we can cancel pi and uh 365 goes into 732 times. So now we get the derivative of V to be two costs. I divided by 365 multiplied by D minus 80, OK, that's the derivative of V. So now we know how our traffic volume is increasing. Now the next question we need to ask ourselves is, when is it increasing at its fastest? In other words, what will make the derivative reach its maximum value? Well, Notice that we have a cosine function in our derivative. And that means then that our derivative is going to be maximum when the cosine function. Is equal to one. OK. And for the cosine function to be equal to one, that means then that we can say. All right, this implies that the cosign of, well. Here, let me, let me write that in a simpler way. This would imply then that our derivative is going to be equal to 2 multiplied by this entire expression being equal to 1, OK? In other words, Then this. Is gonna be equal to 1 and 2 multiplied by 1, and now this is the max, the derivative of, sorry, 2 multiplied by 1 will be equal to 2. So thus, the maximum value, in other words, our volume is going to be increasing the fastest when the traffic volume is increasing at a rate of about 2 cars per day. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Derivative of a Function

Sinusoidal Functions

Critical Points and Maximum Rate of Change

Watch next