The thickness of the ice on a lake for one week is modeled...

The thickness of the ice on a lake for one week is modeled by the function: T(d) = -0.1d^3 + 1.2d^2 - 4.4d + 14.8 where T is the thickness in cm and d is the number of days after December 31st. a) Graph this function using graphing technology. Ensure your axes are properly set up, and state the domain and range that are appropriate for the situation. When do you think the warmest day occurred during the week? Justify your answer, then determine the average rate of change on a short interval near the point you chose. b) Determine the instantaneous rate of change at this point. c) Are the rates you calculated in a) and b) the same or different? Why do you think this is?

Transcript

00:01 So for this problem, we're looking at the thickness of the ice on a lake for a week is modeled for us.

00:05 First, we want to graph the function using graphing technology and give some important values.

00:13 So we see right here, the thickness, we're starting off at 15.

00:19 It's going to look something like this.

00:27 We see that the domain is going to be from zero to...

00:45 About the domain well we know the range is going to be from zero to 15 because those are the possible values you could take over and the domain is going to be from zero to about nine because that's where this intersection occurs for part b we want to know the instantaneous or actually we want to know the warmest day during the week so that's going to be obviously when the thickness is the lowest it could possibly be.

01:16 So the warmest is going to be january 7th because we see that in this portion, that's the last day, and that's when the thickness is lowest.

01:31 So that's about like right here on the graph, right about here is the seventh.

01:37 And that's the lowest point, data point that we have on the graph, given that it's only a week, so from one days to seven days.

01:45 The average rate of change, that's going to be the change in temperature, or that, you know, the change in time over the change in thickness over the change in days.

01:55 So that's going to give us, as we see here, average rate of change is going to be, if we want to choose a short interval, so this is going to be at a given point.

02:27 We want to consider delta t over delta d.

02:32 The average rate of change we see is going to be 0 .4 centimeters.

02:38 But we could do that over any given point...

Question

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