A roast turkey is taken from an oven when its temperature...

A roast turkey is taken from an oven when its temperature has reached $185^{\circ} \mathrm{F}$ and is placed on a table in a room where the temperature is $75^{\circ} \mathrm{F}$ . The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

A roast turkey is taken from an oven when its temperature
has reached $185^{\circ} \mathrm{F}$ and is placed on a table in a room where
the temperature is $75^{\circ} \mathrm{F}$ . The graph shows how the temperature of the turkey decreases and eventually approaches room
temperature. By measuring the slope of the tangent, estimate
the rate of change of the temperature after an hour.

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling is a principle that describes how an object exchanges heat with its surrounding environment, generally stating that the rate of cooling is proportional to the difference between the object’s temperature and the ambient temperature. This law underpins the exponential decay behavior observed and analyzed in the temperature graph.

Graphical Analysis of Functions

Graphical analysis involves interpreting graphs to extract numerical and conceptual information. By examining the temperature versus time graph, one can visually estimate important features such as the rate at which temperature changes by determining the slope of tangents at specific points.

Rate of Change

Rate of change refers to how a quantity changes over time, which in calculus is represented by the derivative. In the context of temperature change, it tells us how fast the temperature is increasing or decreasing at any given moment.

Slope of the Tangent Line

The slope of the tangent line to a curve at a specific point represents the instantaneous rate of change of the quantity depicted by the curve. In this problem, measuring the slope at one hour yields an estimate of how quickly the turkey’s temperature is decreasing at that moment.

Transcript

00:01 All right, we've got a question here.

00:01 We've got a turkey that was originally at 185 degrees.

00:06 It's depicted with this blue line on the graph.

00:09 And you can see that it slowly goes ahead and reduces temperature because apparently you're told in this question that you took the turkey out of the oven and you placed it into a room with the temperature of about 75 degrees fahrenheit.

00:21 So you can see that it slowly, this blue line here, continues to decrease in temperature.

00:28 All right, what we're asked to do here is to estimate the rate of change of the temperature by measuring the slope of the tangent.

00:36 And we're given this line p here that is tangent to our line, the blue line there, which is a representative of the decreasing temperature of the turkey.

00:48 And so what we're supposed to do, what we're asked to do here is to determine the rate of change of the temperature after an hour.

00:55 So what we're going to do is we're going to look at a 60 -minute interval.

01:01 Which would be an hour, and we want to take this line, this red line here, and calculate for its slope.

01:11 We can use that slope to help us determine the rate of change after an hour.

01:17 So if we look at this line p here, we took an interval, we could take it from 30 to 90.

01:25 We can see that at 30, you have the temperature to be approximately, 150, you could say.

01:40 We'll say that that is about 150 because it's right in between 100 and 200.

01:45 And then at 90, we have it to be resting at somewhere close to 100.

01:51 Okay, so what we're going to do is we're going to write out the slope of our tangent, m, and we'll say the slope of our tangent here is when you have the first point of contact, 90, which we know occurs on it intersects with our blue line here at 100...

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