12. On a particular fall day, the outdoor temperature was...

12. On a particular fall day, the outdoor temperature was recorded periodically and a quadratic function was used to model the data. y=f(t)=-0.3t^2 + 8.7t - 2 for 4 ? t ? 22 where t = time of day (in hours) and y = temperature (in degrees) REMARK: Time is measured in hours elapsed since midnight. For instance, t = 4 means 4 am. t = 22 means 10 pm. (a) Using algebraic techniques we have learned, find the maximum temperature predicted by the quadratic model and find the time when it occurred. (b) Find and interpret the average rate of change of f over the interval [5, 9].

Transcript

00:01 So we have an equation that represents the outdoor temperature, and it can be represented this way.

00:08 Y equals f of t, where t is time in elapsed time in hours, and that's going to be equal to negative 0 .3t squared plus 8 .7t minus 2.

00:32 And this is valid for t between 4 and 22 hours elapsed time.

00:44 Again, t is in hours, and y is degrees, temperature, that is.

00:58 The first question is to find the maximum temperature predicted by the quadratic model and the time at which it occurs.

01:06 Well, since this is a quadratic, we note that the line of symmetry for any quadratic, and we do see that this right here is an upside down quadratic, meaning it opens down.

01:20 The vertex for that is going to be at the location minus b over 2a.

01:25 If we think about the standard form of a quadratic, it's y equals a.

01:31 We could do it in x or t is the independent variable, a x squared plus bx plus c, and the line of symmetry is at x.

01:44 Equal minus b over 2a.

01:48 We can use that to find the line in symmetry and that line of symmetry will give us the vertex, that x location right there.

02:02 So in this case, our independent variable is t.

02:05 So this line of symmetry and the vertex is going to occur at t equal to minus b over 2a, which is minus 8 .7 over two times negative 0 .3.

02:23 If you punch it into the calculator, that's going to come out to 14 .5 hours.

02:30 That implies 14 .5 hours t is going to be 30 p .m.

02:42 Because it's 14 and a half hours after midnight...

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