Volume Of A Cube - High School Math

One face of a cube has an area of 64 cm square – What is the volume of this cube?

Answer : 512 cm cubed

In order to calculate the volume of a cube, the length, height and width of the cube must be known since the formula for finding the volume is:
V = l w h
The problem does not provide information about the length, width or height; therefore they must be calculated using the information we were given about the area.

In this problem, we are given the area of one face. We know that the formula for calculating an area for any two dimensional figure is:
A = l w
Since each face of a cube is constructed from a square, then we know that the length and width of this square must be the same and by multiplying the two, the area can be found.

Lets apply what we have:
64 cm square = l w
Since we know that the length and width must be equal for a square, we can take the square root of the area in order to calculate the length of width:

square root of 64 = 8

Therefore, the length and width of each face is 8 cm. Therefore, the height of this cube will also be
8 cm. The volume can now be calculated as follows:

V = (8cm)(8cm)(8cm) = 512 cm cubed.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Reflection Of An Object - Middle School Math

If triangle ABC shown in the figure below is reflected about the y-axis what would the coordinates of the new triangle vertices be?

a. (-3, 3) (-5, 6) (-5, 3)
b. (-3, -3) (-5, -6) (-5, -3)
c. (3, -3) (6, -5) (3, -5)
d. (-3, -3) (-6, -5) (-3, -5)


























Answer: (a) (-3, 3) (-5, 6) (-5, 3)


In order to solve this problem, it is important to understand what happens when an object is reflected about an axis. When an object is reflected about an axis the vertices of the new object will have the same distance from that axis like the original object.

The reflection of triangle ABC is done about the y-axis in this problem. Therefore, the distances each ordered pair has from the y-axis will be the same in the new triangle. This means that the y-coordinate of each ordered pair will be the same in the new triangle like they are in triangle ABC. However, the reflection will mean that the x-coordinates of these points will have the opposite sign in the new object from the original one. In this case, the positive x-coordinates will change to negative x-coordinates.

To simplify this, we’ll look at the location of each ordered pair in the original triangle as follows:

Ordered pair (A) has the original points of (3, 3) therefore the reflected pair will be (-3, 3)

Ordered pair (B) has the original points of (5, 6) therefore the reflected pair will be (-5, 6)

Ordered pair (C) has the original points of (5, 3) therefore the reflected pair will be (-5, 3)

Note in all of the above points, the x-coordinate changed from a positive to a negative while the y-coordinate remained unchanged.

© Copyright 2008.Najwa S. Hirn. All rights reserved.

Elementary Math - Recognizing number sense

While visiting a historic district during a field trip, the students were told the dates that four of the historic houses were built. The chart below list those dates. According to the list, which house was built last?

a. House no. (1)
b. House no. (2)
c. House no. (3)
d. House no. (4)

Answer : ( c ) – House no. (3)

In order to solve this problem, a student must recognize which number in the table represents the highest number. That number will indicate the year the last house was built. By examining the table, it is determined that 1853 is the highest number in the table. Therefore House no. (3) was built last.

The student can list the numbers in ascending or descending order to assist in identifying the highest number.

In order to list the numbers in ascending order, they can be written as follows:

1803 - 1819 - 1836 - 1853

In order to list the numbers in descending order, the can be written as follows:

1853 - 1836 - 1819 - 1803

Both of the above cases indicates that the highest number is 1853 and the lowest number is 1803.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

College Level Math – Vertex, Axis of Symmetry and Maximum/Minimum points

Find the vertex, the axis of symmetry, the maximum or minimum value of the quadratic function.

The above equation is written in the standard quadratic equation form which is:

It is important to be able to calculate the vertex, line of symmetry, and the maximum or minimum points from a quadratic equation. These crucial points can assist in identifying the characteristics of the resulting parabola and thus being able to sketch the resulting graph.

It is important to note that the sign of the (a) term will determine the orientation of the parabola. A positive (a) term – an (a) term greater than zero - will indicate that the parabola will open upwards, while a negative (a) term – an (a) term less than zero – will indicate that the parabola will open downward.

A vertex is either the highest or lowest point of a parabola depending on its orientation. Once calculated, the vertex is written in an (x ,y) ordered pair.

Therefore, it is easy to determine the following characteristics about the parabola presented in this problem. Those characteristics are:

1) This parabola opens upwards since the (a) term is positive or greater than zero.

2) This parabola will have a vertex at its lowest point.

3) This parabola will have a minimum point.

4) The axis of symmetry will be the vertical line passing through the x-coordinate of the vertex and dividing the parabola into two equal parts.

The following steps detail how the vertex is calculated for this problem:

Vertex

Axis of Symmetry

The axis of symmetry is the line that splits the parabola into two identical halves. It passes through the vertex.
Since this quadratic opens upwards, that means that the axis of symmetry is the x co-ordinate of the vertex. Therefore, the axis of symmetry is the vertical line that passes through x = 5

Maximum or Minimum points

The vertex of a parabola is usually considered as either a maximum or a Minimum value.

We determine if the point is a maximum or minimum by looking at the a co-efficient of the a-term of the quadratic equation. Since the a-term in this quadratic equation is greater than zero, then the vertex is a minimum point because the parabola opens upwards.

Therefore for this quadratic the minimum value point occurs at (5,-28)
If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to the y-co-ordinate of the vertex. That means the minimum point for this quadratic = -28
This minimum value occurs at x = = -b/2a which is the x-co-ordinate of the vertex. For this quadratic the minimum value = 5



© Copyright 2008.Najwa S. Hirn. All rights reserved.