Spherical Zone Problems

Category: Solid Geometry, Algebra

"Published in Newark, California, USA"

Show that the area of a zone of one base is equal to the area of the circle whose radius is the chord c of the generating arc AB of the zone. 

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Solution:

A spherical zone or zone, in short is a portion of the surface of a sphere from its circular cross section to its end (for one base) or between two parallel circular planes (for two bases).  The above figure is a zone of one base.

In this problem, we will compare the area of a zone with the area of a circle whose radius is chord AB or c. Let's see if they will be equal or not. Analyze and label further the given figure above as follows

Photo by Math Principles in Everyday Life

Apply Pythagorean Theorem for ∆OCB,

Apply Pythagorean Theorem for ∆ABC,

The area of a Zone is equal to

The area of a circle whose radius is chord AB or c is equal to

Therefore,

                                       Area of Zone = Area of Circle

Equation - Perpendicular Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a line that passes through the point (2, -1) and perpendicular to the line 2x - 3y + 4 = 0.

Solution:

Consider the given line

Rewrite the equation of a line in slope-intercept form as follows

The slope of a line is m₁ = ²/₃ and the y-intercept is b = ⁴/₃. To draw or sketch a line, plot  ⁴/₃ at the y-axis. This is your first point of the line at (0, ⁴/₃). Next, use the slope to get the second point. From the first point, count 3 units to the right and then 2 units upward. Connect the two points and you have now a line as follows

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Finally, plot (2, -1) and draw a line perpendicular to the given line that contains a given point as follows

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If two lines are perpendicular, then their slopes are negative reciprocals to each other which means that m₂ = -¹/_m1 = ^-3/₂. Therefore, using the point-slope form, the equation of another line that passes through the point (2, -1) is

Equation - Parallel Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a line that passes through the point (2, -1) and parallel to the line 2x - 3y + 4 = 0.

Solution:

Consider the given line

Rewrite the equation of a line in slope-intercept form as follows

The slope of a line is m₁ = ²/₃ and the y-intercept is b = ⁴/₃. To draw or sketch a line, plot  ⁴/₃ at the y-axis. This is your first point of the line at (0, ⁴/₃). Next, use the slope to get the second point. From the first point, count 3 units to the right and then 2 units upward. Connect the two points and you have now a line as follows

Photo by Math Principles in Everyday Life

Finally, plot (2, -1) and draw a line parallel to the given line that contains a given point as follows

Photo by Math Principles in Everyday Life

If two lines are parallel, then their slopes are equal, which is m₁ = m₂ = ²/₃. Therefore, using the point-slope form, the equation of another line that passes through the point (2, -1) is