## Thursday, January 31, 2013

### Maximum Minimum Problem, 2

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

The cost of fuel per hour for running a ship is proportional to the cube of the speed and is \$27 per hour when the speed is 12 miles per hour. Other costs amount to \$128 per hour regardless of the speed. Express the cost per mile as a function of the speed, and find the speed that makes this cost a minimum.

Solution:

The first thing that we have to do is to analyze the given word problem as follows

Let C = cost of fuel per hour
v = speed of a ship

From the word problem, "The cost of fuel per hour for running a ship is proportional to the cube of the speed", the working equation will be

If C = \$27 per hour
v = 12 miles per hour

then the value of k will be

Therefore

From the word problem, "Other costs amount to \$128 per hour regardless of the speed", the final working equation will be

To get the minimum cost of a fuel as a function of the speed,  take the derivative of the above equation with respect to v as follows

Set dC/dv = 0 since we are getting the minimum cost of a fuel as follows

## Tuesday, January 29, 2013

### Solving Trigonometric Equations, 3

Category: Trigonometry

"Published in Newark, California, USA"

Solve for the unknown angle for

Solution:

Consider the given equation

As you notice that the coefficients of Sin 2θ and Cos 2θ are not equal. If you will expand the above equation in order to convert the double angles into single angles, the above equation will be more complicated and it will be hard to simplify and solve the equation. Don't worry, we have a solution or technique to solve this kind of trigonometric equation.

Draw a right triangle to represent the coefficients of Sin 2θ and Cos 2θ. The coefficient of Sin 2θ will be the adjacent side of a right triangle. The coefficient of Cos 2θ will be the opposite side of a right triangle. Solve for the hypotenuse and get the trigonometric functions as follows Photo by Math Principles in Everyday Life

From the given equation

Divide both sides of the equation by the hypotenuse which is 2 as follows

Substitute the coefficients of Sin 2θ and Cos 2θ by their equivalent trigonometric functions as follows

but

The above equation becomes

where n = number of revolutions.