Two Intersecting Lines, 3

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a line that passes through the intersection of x - y = 0 and 3x - 2y = 2 and forms a triangle at 1st Quadrant whose area is 9.

Solution:

To illustrate the problem, it is better to sketch the graph of two lines as follows:

For x - y = 0,

                    x - y = 0
                    y = x

                    slope (∆y/∆x), m = 1
                    y-intercept = 0

To trace the graph, plot 0 at the y-axis. This is your first point of the line (0, 0). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 1 unit upward.

For 3x - 2y = 2, 

                    3x - 2y = 2
                    2y = 3x - 2
                    y = 3/2 x - 1

                    slope (∆y/∆x), m = 3/2
                    y-intercept = -1

To trace the graph, plot -1 at the y-axis. This is your first point of the line (0, -1). Next, use the slope to get the second point. From the first point, count 2 units to the right and then 3 units upward. 

Photo by Math Principles in Everyday Life

  
To get their point of intersection, we have to use the two given equations and solve for x and y as follows

from

Substitute the second equation to the first equation, we have

Substitute x to either of the two equations,

Therefore, their point of intersection is P(2, 2).

Photo by Math Principles in Everyday Life

From the given word problem says "...forms a triangle at 1st Quadrant whose area is 9.", then the line that passes through point P will intersect at x-axis and y-axis. The resulting figure is a right triangle whose sides are x and y. The right angle is the origin while point P is located at the hypotenuse as follows

Photo by Math Principles in Everyday Life

By using the formula for the area of a triangle, the first equation will be

By using the formula for getting the slope of two points, the second equation will be

Substitute the value of y from the first equation to the second equation, we have

Multiply both sides of the equation by x

If you will choose the positive sign,

Substitute the value of x either to the first equation or second equation, we have

The slope of a line is calculated as follows

Therefore, using the Point-Slope Form, the equation of a line is

If you will choose the negative sign,

Substitute the value of x either to the first equation or second equation, we have

The slope of a line is calculated as follows

Therefore, using the Point-Slope Form, the equation of a line is

 

Approximation - Error Problem, 3

Category: Differential Calculus, Solid Geometry

"Published in Newark, California, USA"

Considering the volume of a spherical shell as an increment of volume of a sphere, find approximately the volume of a spherical shell whose outer diameter is 8 inches and whose thickness is 1/16 inch.

Solution:

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

Photo by Math Principles in Everyday Life

If the diameter of a sphere is given in the problem, then the volume of a sphere is

but D = 2R

Since the thickness of a sphere is given and we want to find the volume of a spherical shell, then we have to take the differentials on both sides of the equation. The differential of a volume of a sphere is the same as the volume of a spherical shell. 

Substitute the value of R which is ½ D or 4 in. and dR which is 1/16 in. to the above equation, we have

Therefore, the final answer is

Word Problem - Distance Problem

Category: Algebra

"Published in Suisun City, California, USA"

The fuel consumption for William's car is 30 mi/gal on the highway and 25 mi/gal in the city. On vacation trip of 400 miles, he used 14 gallons of gasoline. How many highway miles and city miles did he drive on this trip?

Solution: 

The given word problem above is about getting the trip distances of highway and city miles that involves the principles of solving two equations, two unkowns because the two fuel consumption for William's car are given. Also, the total vacation trip miles and total gallons of gasoline are given in the problem. 

Let x = be the total trip distance of highway miles
      y = be the total trip distance of city miles
      30 mi/gal = fuel consuption of highway miles
      25 mi/gal = fuel consuption of city miles

If the total vacation trip miles is 400 miles, then we can write the first equation as follows 

If the total gallons of gasoline is 14 gallons, then we can write the second equation as follows

Multiply both sides of the equation by their Least Common Denominator (LCD) which is 150 as follows

The two unknowns of two linear equations can be solved by elimination method. Consider the two linear equations as follows

Multiply the first equation by 5 and -1 at the second equation, we have

                                                    
When you add the two equations, x will be eliminated as follows

Substitute the value of y either of the two equations above as follows

Therefore, the final answers are

        Total Trip Distance of Highway Miles = 300 miles

        Total Trip Distance of City Miles = 100 miles