Differentiation - Rate Problem

Category: Differential Calculus, Plane Geometry, Trigonometry

"Published in Newark, California, USA"

Each of two sides of a triangle are increasing at the rate of ½ foot per second, and the included angle is decreasing at 2° per second. Find the rate of change of the area when the sides and included angle are respectively 5 feet, 8 feet, and 60°.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

From Plane Geometry, the area of a triangle is given by the formula

From Trigonometry, we know that

Therefore,

Take the derivative on both sides of the equation with respect to time, we have

but   b = 8 feet
        c = 5 feet
        θ = 60°
        ^db/_dt = ^dc/_dt = ½ ft/sec
        ^dθ/_dt = - 2°/sec (negative because decreases)
                = (- 2°/sec) x (π/180°) = - π/90 radians/sec

Substitute the values to the above equation, we have

Therefore,

Since the rate is positive, then the area is increasing.

More Integration Procedures, 4

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:  

The above equation cannot be integrated by a simple integration. In this type of equation, we will use the reciprocal substitution in order to simplify the equation and then integrate by a simple integration. Let's consider the above equation

Let 

so that

Substitute the values of x and dx to the above equation, we have

but

Therefore,

or

Newtons Law - Cooling, 2

Category: Chemical Engineering Math, Differential Equations, Algebra

"Published in Newark, California, USA"

The body of a murder victim was discovered at 11:00 pm. At 11:30 pm, the victim's body temperature was measured to be 94.6 °F. After 1 hour, the body temperature was 93.4 °F. The room where the body was found at a constant temperature of 70 °F. Assuming that Newton's Law of Cooling is applicable and assuming that the normal human body temperature is 37 °C, determine the time of death.

Solution:

This is a great application of Newton's Law of Cooling. Mostly the Federal Bureau of Investigation (FBI) and the police officers in United States of America are using this method to calculate the time of a death or murder. Even in other countries like Philippines, Canada, Mexico, Japan, Italy, China, and so on are using this method, too. Let's analyze the given word problem as follows

Let u = be the temperature of a dead body
      t = be the time of death
      k = constant of cooling/heating

According to Newton's Law, the time rate of change of temperature is proportional to the temperature difference. 
 

      
The value of k is negative because it is a cooling process. When k is positive, then it is a heating process. The temperature of the surrounding is always a constant which is 70 °F. Solve for the general solution of the above equation, we have
 

     
Integrate on both sides of the equation
 

 

       
Take the inverse natural logarithm on both sides of the equation
 

 

       
but e^C is still a constant. The above equation becomes
 

       
To solve for the values of k and C, we need to get the values of the limits as follows

If u₁ = 94.6 °F, then t₁ = 0 (measured at 11:30 pm)
   u₂ = 93.4 °F, then t₂ = 1.0 hr (measured at 12:30 am)

Substitute the first limit to the above equation to solve for the value of C
 

 

 

 
Therefore,
 

Substitute the second limit to the above equation to solve for the value of k

 

 
Take natural logarithm on both sides of the equation

Therefore,
 

   
Next, we need to use the normal temperature of a body (u = 37 °C = 98.6 °F) in order to calculate the time of death of a person as follows
    

   

Take natural logarithm on both sides of the equation
             

              

or

Since the value of time is negative, then we have to subtract it from the time where a dead body was first measured, which is at 11:30 pm, as follows

         11:30        ------>  11:30:00    ------->   11:29:60
         - 3:00:48              - 3:00:48                - 3:00:48
                                                                ---------------
                                                                     8:29:12

Therefore, a person is dead at 8:29:12 pm.