Problem 457 - Calculus Volume 1

Question

For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 \(\mathrm{m}\) and width 388 m. Assume the current depth of the water is 180 \(\mathrm{m} .\) The density of water is 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and \(45^{\circ} \mathrm{F}\) outside and the temperature of the body is \(78^{\circ} \mathrm{F}\) , You know the cooling constant is \(k=0.008244 \mathrm{F} / \mathrm{min}\) . When did the victim die, assuming that a human's temperature is \(98^{\circ} \mathrm{F} ?\)

Step-by-Step Solution

Answer

The victim died approximately 5 hours and 16 minutes before noon, around 6:44 AM.

Step 1: Understanding the Problem

To determine the time of death, use the formula for Newton's Law of Cooling: \[T(t) = T_{ ext{env}} + (T_0 - T_{ ext{env}}) imes e^{-kt}\]where:- \( T(t) \) is the body temperature at time \( t \).- \( T_{ ext{env}} \) is the environmental temperature, which is \( 45^{\circ} \mathrm{F} \).- \( T_0 \) is the initial body temperature, \( 98^{\circ} \mathrm{F} \).- \( k \) is the cooling constant, \( 0.008244 \mathrm{F}/\mathrm{min} \).- \( t \) is the time passed since death.

Step 2: Initial Setup

Substitute the known values into the formula:\[78 = 45 + (98 - 45) \times e^{-0.008244t}\]This equation will help us find the time since the victim died (\( t \)).

Step 3: Simplify the Equation

Subtract \( 45 \) from both sides to isolate the exponential term:\[33 = 53 \times e^{-0.008244t}\]

Step 4: Solve for Exponential Term

Divide both sides by \( 53 \):\[\frac{33}{53} = e^{-0.008244t}\]

Step 5: Apply Logarithm

Take the natural logarithm of both sides to solve for \( t \):\[\ln\left(\frac{33}{53}\right) = -0.008244t\]

Step 6: Calculate Time \, t

Rearranging for \( t \), we have:\[t = \frac{\ln\left(\frac{33}{53}\right)}{-0.008244}\]Evaluate this expression using a calculator.

Step 7: Compute and Interpret Result

Upon computation, \( t \approx 316 \) minutes. This means it has been approximately 316 minutes since the victim died. Convert this into hours and minutes: \( 316 \) minutes is 5 hours and 16 minutes.

Key Concepts

Exponential Decay

Newton's Law of Cooling beautifully models the process of cooling through exponential decay. The term **exponential decay** is used to describe a pattern where the rate of decrease is proportional to the current value. With respect to cooling, this means that the rate at which something cools is directly related to how much warmer it is than its surroundings.

In the derived formula:

• \(T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) \times e^{-kt}\),

\(e^{-kt}\) is the exponential decay factor. Its presence indicates how rapidly the temperature approaches the environmental temperature (steady-state). If we picture this as a graph, the body temperature slowly decreases with each passing unit of time, approaching but never reaching complete equilibrium instantaneously.

**Key Takeaways:**
- Exponential decay lets us see how quickly the process slows down.
- The term helps us determine the cooling rate efficiently by calculating how temperature differences shrink over time with a constant environment.

Temperature Gradient

Another crucial aspect when using Newton's Law of Cooling is the **temperature gradient**. This refers to the difference in temperature between the object cooling down and its environment.

This difference determines the rate at which heat flows from the object to the surroundings, directly impacting the speed of cooling. In our formula, this is represented by the component \((T_0 - T_{\text{env}})\).

The significance of this gradient is easy to observe in everyday scenarios. If you have a cup of hot coffee in a cold room, the temperature difference is large, hence the cooling process is rapid initially. As the coffee cools and its temperature decreases, so does that gradient, resulting in slower cooling.

**Important Points to Remember:**

• A larger temperature gradient means faster cooling.
• As the object approaches the temperature of the environment, the gradient decreases, slowing the cooling pace.

Cooling Constant

The role of the **cooling constant**, represented as \(k\), is another fundamental element in Newton's Law of Cooling. This constant specifically determines how quickly an object will lose its heat relative to the temperature difference between the object and its environment.

The value of \(k\) is determined based on experimental factors and depends on properties like the shape, size, and surface area of the object, plus environmental conditions.

• In our scenario, \(k = 0.008244 \text{ F/min}\), reflects the specific rates of cooling peculiar to the body and the conditions.
  

With a higher value of \(k\), you can expect rapid cooling. Conversely, a smaller \(k\) means the cooling will occur at a more gradual rate.

**Why It Matters:**
- Allows precise predictions of cooling over time.
- Helps scientific calculations remain relevant by describing a multitude of scenarios, be it a cooling drink or a forensic investigation.