Right Circular Cylinder - Cube

Category: Solid Geometry

"Published in Newark, California, USA"

Find the volume and total area of the largest cube of wood that can be cut from a log of circular cross section whose radius is 12.7 in.

Photo by Math Principles in Everyday Life

Solution:

Since the radius of a wood log is given, we have to consider the cross section of a wood log as follows

Photo by Math Principles in Everyday Life

As you notice that a square is inscribed in a circle. There are 6 equal squares in a cube and so all the sides of a cube are equal. The radius of a wood log or a circle is equal to ½ of its diagonal. The diagonals of a square bisect each other at a right angle. A right triangle that has two equal sides and two equal angles is called an isosceles right triangle. In this figure, there are four isosceles right triangles in a square. By Pythagorean Theorem,

The length of a side of a cube is 17.9605 in.

The total area of a cube is

The volume of a cube is 

Approximate Value - Cube Root

Category: Differential Calculus, Arithmetic

"Published in Newark, California, USA"

Find the approximate value of ∛200 without using a calculator.

Solution:

Well, we know that

             1³ = 1

             2³ = 8

             3³ = 27

             4³ = 64

             5³ = 125

             6³ = 216       and so on.

Since 200 is between 125 and 216, the ∛200  is between 5 and 6. We can get the approximate value of ∛200  without using a calculator as follows

Let

then

If   x = 216 (please choose this number because it is close to 200)
   dx = 200 - 216 = -16

then dy will be

Therefore

using a calculator

and using a scientific calculator

which is close to the final answer.

Word Problem - Number Problem

Category: Algebra

"Published in Newark, California, USA"

Find a 2-digit number whose ten's digit is 3 more than the unit's digit. The sum of the 2 digits is 7.

Solution:

Well, the given word problem is about a number problem where you will find the value of ten's and one's digits. Let's analyze the word problem as follows

Let        x = value of one's digit
       x + 3 = value of ten's digit

From the word statement, "the sum of the 2 digits is 7", the working equation for the given problem is

                                        x + x + 3 = 7

                                            2x + 3 = 7

                                                  2x = 7 - 3

                                                  2x = 4

                                                    x = 2

One's digit = x = 2
Ten's digit = x + 3 = 2 + 3 = 5

Therefore, the number is 52. The sum of the digits is 5 + 2 = 7 which is correct.