Circular Cylinder Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A pine log 40 ft. long has a uniform circular cross section of radius 11 in. The ends lie in parallel planes which are not perpendicular to the axis of the log. How much paint will be required to give the log (ends not included) two coats, if 1 gal. of paint covers 800 sq. ft. of surface? If pine weighs 31 lbs. per cu. ft., find the weight of the log.

Solution:

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

Photo by Math Principles in Everyday Life

The surface or lateral area of a circular cylinder which is a pine log is

where p_k is the circumference or perimeter of the cross section and e is the length of an element or the slant height. Substitute the values, we have

Therefore, the amount of paint required to paint the surface of a pine logs in two coats is

The volume of a circular cylinder which is a pine log is

where K is the area of a cross section and e is the length of an element or the slant height. Substitute the values, we have
 

 

 

 

 

Therefore, the weight of a pine log is
 

 

Trapezoid and Quadrilateral Problems, 7

Category: Plane Geometry

"Published in Newark, California, USA"

Two congruent angles of an isosceles trapezoid have measures 3x + 10 and 5x - 10. Find the value of x and then give the measures of all angles of the trapezoid.

Solution:

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

Photo by Math Principles in Everyday Life

Isosceles trapezoid is a trapezoid whose two opposite sides and two adjacent angles are congruent. If the two adjacent angles of the lower base are congruent, then the two adjacent angles of the upper base are also congruent.
 
For any trapezoid, the two adjacent sides of a leg are supplementary angles because the upper and lower bases are parallel. 

If that's the case, the value of x is

 

 

Hence, the measurement of an angle at the lower base of an isosceles trapezoid is
 

 

 

and at the upper base is
 

 

 

Therefore, the measurements of the two congruent angles of an isosceles trapezoid are 77.5° and 102.5°.
 

Trapezoid and Quadrilateral Problems, 6

Category: Plane Geometry

"Published in Newark, California, USA"

One angle of an isosceles trapezoid has measure 57°. Find the measures of the other angles.

Solution:

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

Photo by Math Principles in Everyday Life

Isosceles trapezoid is a trapezoid whose two opposite sides and two adjacent angles are congruent. If the two adjacent angles of the lower base are congruent, then the two adjacent angles of the upper base are also congruent.
 
For any trapezoid, the two adjacent sides of a leg are supplementary angles because the upper and lower bases are parallel. 
 
Therefore, the other angles of an isosceles trapezoid are
 

 

 

 
There's another way in solving for the other angles of an isosceles trapezoid. For any four closed figures like quadrilateral, trapezoid, parallelogram, rhombus, rectangle, and square, the sum of their internal angles is equal to 360°. 
 
Therefore, the other angles of an isosceles trapezoid are
 

 

 

 

 

Trapezoid and Quadrilateral Problems, 5

Category: Plane Geometry

"Published in Newark, California, USA"

Given a trapezoid with its median. Find the value of x.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above 

Photo by Math Principles in Everyday Life

If a theorem says "The median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths.", then the value of x, therefore is

 

Trapezoid and Quadrilateral Problems, 4

Category: Plane Geometry

"Published in Newark, California, USA"

Given a trapezoid with its median. Find the value of x.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

If a theorem says "The median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths.", then the value of x, therefore is

 

Parallel Lines Problems

Category: Plane Geometry

"Published in Newark, California, USA"

Two lines are cut by four parallel lines of equal distance as shown in the figure. Find the values of x and y.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

If a theorem says "If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.", then we can equate the two line segments in order to solve for the value of x as follows

Finally, equate the other two line segments in order to solve for the value of y as follows