Tangent Lines and Normal Lines, 5

Category: Differential Calculus

"Published in Vacaville, California, USA"

Show that the curvehas no tangent line with slope 4.

Solution:

Consider the given curve

As we know that the slope of any curve is equal to the first derivative of the equation of any curve with respect to the independent variable which is x in most cases. Take the derivative of the given equation with respect to x, we have

If the slope of a curve is dy/dx = 4 which is also the slope of a tangent line, then the above equation becomes

From the resulting equation, the values of x will be the x values of the intersection of a curve and a tangent line. By using quadratic formula, the values of x are

Since the values of x are imaginary numbers or complex numbers, then there's no tangent line for the given curve. The given curve did not intersect with the tangent line of a given slope.

Maximum and Minimum Problems, 9

Category: Differential Calculus

"Published in Vacaville, California, USA"

A rain gutter is to be constructed from a metal sheet of width 30 cm. by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?

Photo by Math Principles in Everyday Life

Solution:

From the given figure, it is a cross section of a gutter in a form of isosceles trapezoid because the opposite sides as well as the opposite angles of the lower base are congruent. 

In the given word problem, maximum amount means maximum volume. Since the given figure is a cross section and the height of a gutter is not given, then it is a maximum area.  

If the length of the opposite sides of a trapezoid as well as the opposite angles at the lower base are given, then we can solve for the altitude and the length of the upper base. 

Photo by Math Principles in Everyday Life

Since the two opposite triangles at the opposite sides of a trapezoid are right triangles, then we can use the trigonometric functions as follows

The area of a cross section of a gutter which is an isosceles trapezoid is

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

Since we want to get the maximum amount of water or maximum area of a cross section of a gutter, then set dA/dθ, we have

             
but
 

 

then the above equation becomes
 

 

 

 

 

Equate each factors to zero and solve for the value of θ.
 

 

 

 

Since the value of θ is a straight angle, then we cannot accept this as an answer. Let's consider the other factor and solve for the value of θ, we have
 

 

 

 

Since the value of θ is an acute angle, then we can consider this one as an answer.
 

Index of Refraction Problems

Category: Physics

"Published in Newark, California, USA"

A ray of light strikes a water surface at an angle of incidence of 40°. What is the angle of refraction in the water? 

Solution:

The given problem is about refraction of light in which the speed of light will change as it passes through the material. There's a bending of ray of light also in the material. To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

According to Snell's Law, the index of refraction is inversely proportional to angle of refraction. The higher the value of index of refraction, the smaller the angle of refraction is. 

From the Table of Index of Refraction, the index of refraction of air at 20°C is 1.000 and water at 20°C is 1.333. Therefore, the angle of refraction in the water is

Take the inverse sine on both sides of the equation, we have

                               or