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Question

3. A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function $M$ models the temperature of the milk at time $t$, where $M(t)$ is measured in degrees Celsius $\left({ }^{\circ} \mathrm{C}\right)$ and $t$ is the number of minutes since the bottle was placed in the pan. $M$ satisfies the differential equation $\frac{d M}{d t}=\frac{1}{4}(40-M)$. At time $t=0$, the temperature of the milk is $5^{\circ} \mathrm{C}$. It can be shown that $M(t)<40$ for all values of $t$. (a) A slope field for the differential equation $\frac{d M}{d t}=\frac{1}{4}(40-M)$ is shown. Sketch the solution curve through the point $(0,5)$.

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Benjamin

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Solution By Steps

Step 1: Initial Condition

Given

M(0) = 5.

Step 2: Differential Equation

The given differential equation is

\frac{dM}{dt} = \frac{1}{4}(40 - M).

Step 3: Slope at

(0,5)

Substitute

M = 5 into the differential equation to find the slope at

(0,5).

Step 4: Slope Field Interpretation

In the slope field, at

(0,5), the slope indicates the direction of the solution curve at that point.

Step 5: Sketching the Solution Curve

Follow the direction indicated by the slope field at

(0,5) to sketch the solution curve.

Final Answer

Sketch the solution curve starting at

(0,5) following the direction indicated by the slope field.

Key Concept

Slope Field Interpretation

Key Concept Explanation

Slope fields provide visual guidance on the direction of solution curves for differential equations at specific points. By following the slopes at given points, one can sketch the solution curve accurately.

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