Volume - Cube, Cutting Plane

Category: Solid Geometry

"Published in Newark, California, USA"

The plane section ABCD shown in the figure is cut from a cube of edge a. Find the area of this section if D and C are each at the midpoint of an edge.

Photo by Math Principles in Everyday Life

Solution:

The given problem above is asking for the area of a cutting plane ABCD. If C and D are the midpoints of the parallel edges of a cube, then the parallel edges of a cube will bisect into equal parts as shown in the figure below

Photo by Math Principles in Everyday Life

As you notice that there are two right triangles in a cube that are parallel to each other. We need to solve for their hypotenuse using Pythagorean Theorem, as follows

Since AD ║ BC, AB ║ CD, AD ≅ BC, and AB ≅ CD, then quadrilateral ABCD is a rectangle which is a cutting plane. Therefore, the area of a rectangle is

Similar Triangles

Category: Plane Geometry

"Published in Newark, California, USA"

The points C and D in the figure lie on level ground in the same vertical plane with the tip B of the tower AB. If the tower AB is 300 ft. high and measurements give A₁B₁ = 5 ft., CA₁ = 12 ft., A₂B₂ = 6 ft., and A₂D = 8 ft., find the distance CD.

Photo by Math Principles in Everyday Life

Solution:

The given problem above is asking for a distance of two points which is CD using similar triangles. The tower is located between CD and assuming that it is perpendicular to the ground. In this case, there are four right triangles in the figure. Label further the figure above, we have

Photo by Math Principles in Everyday Life

Consider ∆ABC:

Using similar triangles, we have

Consider ∆ABD:

Using similar triangles, we have

Therefore,

Variable Separation, 3

Category: Differential Equations, Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Find the general solution for

Solution:

Consider the given equation above

The given equation above is a differential equation because it contains the differentials like dz and dt. We have to eliminate the differentials using the techniques of integration as follows

Using the separation of variables

Integrate on both sides of the equation

or

Multiply both sides of the equation by e^z, we have