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}F $ and is placed on a table in a room where the temperature is $ 75^{\circ}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

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.

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.

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.

Transcript

00:01 Hello and welcome to this video solution of numerate so here it's given that a rose turkey is taken from an oven when its temperature has reached 185 degree fahrenheit right so i can see from this graph here this black curve has started from 185 degree fahrenheit right at t equal to 0 right at it starts at origin if you see the graph and it's placed on a table in a room where the temperature is 75 degree fahrenheit right so eventually it will reach the room temperature that we also see here by measuring the slope of the tangent you have to estimate the rate of change of the temperature after an hour right so this point p is at 60 minutes which is showing a temperature of 125 degree fahrenheit right so first of all what you can do is you can apply newton's law of cooling right and this to calculate the temperature after any time interval right but here what we need to do is you need to calculate the rate of change of temperature after an hour right so for that what you need to do is you need to find the slope? of this tangent at this point and with that what you can do is you can find adjacent points where? we can find that time and temperatures right so somewhere around let's say this is at this point you have caught this is 90 -minute right the temperature is around a bit less than 150 which is we can write like 145 right 145 degree fahrenheit this point one and you have got point two somewhere around here and this is sort of 90 minutes and this temperature is a bit above hundred right which is let's 110 right so between this so what i can do is the point one is you've got 30 and 145 and this point to where you have got 90 and 110 right so you can use these two points to calculate the slope so what we need to do is dt over dt this rate of change of temperature with time you have to find and the final point is 110 minus the initial is 145 the final time is 90 and initial time is 30 right and this is equal to let's see how much we have here now this is minus 35 over 60 right around minus 0 .58 degree fahrenheit per minute right is a rate of change of temperature...

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