A body starts cooling in a surrounding which is at a temperature of θ0-A graph is between temper

step 1 This question is about that a body starts cooling in a surrounding which is at a temperature of

A body starts cooling in a surrounding which is at a...

A body starts cooling in a surrounding which is at a temperature of $\theta_{0}-\mathrm{A}$ graph is between temperature and time $t$. Newton's cooling is Tangents are the curve at and $Q$ at temperatures of $\quad \theta_{2}$ respectively. If tangent meet the time axis at angle of $\phi_{2}$ and $\phi_{1}$, then : (a) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{1}-\theta_{0}}{\theta_{2}-\theta_{0}}$ (b) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{2}-\theta_{0}}{\theta_{1}-\theta_{0}}$ (c) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{1}}{\theta_{2}}$ (d) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{2}}{\theta_{1}}$

A body starts cooling in a surrounding which is at a temperature of $\theta_{0}-\mathrm{A}$ graph is between temperature and time $t$. Newton's cooling is Tangents are the curve at and $Q$ at temperatures of $\quad \theta_{2}$
respectively. If tangent meet the time axis at angle of $\phi_{2}$ and $\phi_{1}$, then :
(a) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{1}-\theta_{0}}{\theta_{2}-\theta_{0}}$
(b) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{2}-\theta_{0}}{\theta_{1}-\theta_{0}}$
(c) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{1}}{\theta_{2}}$
(d) $\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{2}}{\theta_{1}}$

Key Concepts

Proportional Relationships Using Tangent Slopes

The problem involves ratios of the tangent of angles at different points on the cooling curve, which correspond to proportions of the slopes (rates of temperature change) at those points. Understanding these proportional relationships helps in establishing connections between the rate of cooling and the temperature differences from the ambient temperature.

Slope of a Curve

The slope of the tangent to the cooling curve represents the derivative of the temperature with respect to time at a specific point. This concept is key for analyzing how fast the temperature is changing, which is inherent in dynamic processes like cooling or heating.

Angle of Tangency with the Time Axis

The angle that a tangent line makes with the time axis is directly related to the slope of the tangent because the tangent of this angle equals the slope. Therefore, analyzing the tangent of the angle provides a geometric interpretation of the rate of temperature change.

Newton's Law of Cooling

Newton's Law of Cooling describes the rate at which an object changes temperature through radiation or conduction relative to the temperature of its surrounding environment. This relationship is typically expressed as a first?order differential equation, where the rate of change of temperature is proportional to the difference between the object's temperature and the ambient temperature.

Tangent Line on a Temperature-Time Graph

In the context of a temperature versus time graph, the tangent line at a specific point represents the instantaneous rate of change of temperature. The point of tangency portrays the state of the system at a given moment and the slope of the tangent provides insights into the cooling rate at that precise point in time.

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