Circle - Square Problems

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the area of the largest circle which can be cut from a square of edge 4 in. What is the area of the material wasted?

Solution:

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

Photo by Math Principles in Everyday Life

Area of a square:

The edge of a square is equal to the diameter of a circle as shown in the figure above. A circle is tangent to the sides of a square. If the edge of a square is 4 in., then it follows that the diameter of a circle is 4 in. If the diameter of a circle is 4 in., then the radius of a circle is 2 in.

Area of a circle:

or

Therefore, the area of material wasted is

 or

 

Square, Rectangle, Parallelogram Problems, 6

Category: Plane Geometry

"Published in Newark, California, USA"

A window glass is 4 ft. 2 in. by 2 ft. 10 in. Find its area.

Solution:

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

Photo by Math Principles in Everyday Life

We notice that the dimensions are not the same because the given units are in feet and inches. Let's convert first all the dimensions in feet only as follows

For the length of a rectangle:
 

 

 

 

 

   
For the width of a rectangle:
 

 

 

 

 

   
Therefore, the area of a rectangle which is a window glass is
 

 

 

 

 

or

   
You can also express your answer in terms of square inches as follows
 

 

 

Finding Equation - Curve, 14

Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of the curve for which y" = 2, and which has a slope of -2 at its point of inflection (1, 3).

Solution:

The concavity of a curve is equal to the second derivative of a curve with respect to x. In this case, y" = d²y/ dx². Let's consider the given concavity of a curve

We can rewrite the above equation as follows

Multiply both sides of the equation by dx, we have 

Integrate on both sides of the equation, we have 

The point of inflection is a point where the direction of the concavity of a curve will start to change. In this case (1, 3) is the point of inflection of a curve. Since it is also included in the curve, then we can use it to substitute the value of x and y later in the problem.

Substitute the value of the given slope and the point of inflection to the above equation in order to solve for the value of a constant, we have

Hence, the above equation becomes 

Multiply both sides of the equation by dx, we have 

Integrate on both sides of the equation, we have 

In order to solve for the value of a constant, substitute the value of x and y from the coordinates of a given point of inflection to the above equation, we have 

Therefore, the equation of a curve is