Area of the Square: Increasing a specific element by addition of.....or multiplication by.......

Let's denote the initial cube's edge length as x,

The formula for the volume of a cube with edge length b is:

$V=b^3$

Therefore the volume of the initial cube (meaning before increasing its edge) is:

$V_1=x^3$

Proceed to increase the cube's edge by a factor of 6, meaning the edge length is now: 6x . Therefore the volume of the new cube is:

$V_2=(6x)^3=6^3x^3$

In the second step we simplified the expression for the new cube's volume by using the power rule for multiplication in parentheses:

$(z\cdot y)^n=z^n\cdot y^n$

We applied the power to each term inside of the parentheses multiplication.

Next we'll answer the question that was asked - "By what factor did the cube's volume increase", meaning - by what factor do we multiply the old cube's volume (before increasing its edge) to obtain the new cube's volume?

Therefore to answer this question we simply divide the new cube's volume by the old cube's volume:

$\frac{V_2}{V_1}=\frac{6^3x^3}{x^3}=6^3$

In the first step we substituted the expressions for the volumes of the old and new cubes that we obtained above. In the second step we reduced the common factor between the numerator and denominator,

Therefore we understood that the cube's volume increased by a factor of -$6^3$when we increased its edge by a factor of 6,

The correct answer is b.

To solve this problem, let's follow these steps:

• Step 1: Determine the side length of the new square.
• Step 2: Calculate the area of the new square using the area formula for a square.

Now, let's work through the solution:

Step 1: Calculate the side length of the new square.

The side length of the original square is 5 units. The problem states that the side of the new square is longer by 5 units than the original square. Therefore, the side length of the new square is:

$\text{New Side Length} = 5 + 5 = 10 \text{ units}$

Step 2: Calculate the area of the new square.

To find the area of the new square, we use the formula for the area of a square, which is the side length squared:

$\text{Area of the New Square} = (\text{Side Length})^2 = 10^2 = 100 \text{ square units}$

Therefore, the area of the new square is 100 square units.

Thus, the correct answer is option 3: 100.

To solve this problem, we'll follow these steps:

• Step 1: Identify the original side length.
• Step 2: Determine the new side length by subtracting 3 cm from the original side length.
• Step 3: Apply the formula for the area of a square to find the area of the new square.

Now, let's work through each step:
Step 1: The original square has sides measuring 6 cm.
Step 2: The new square has sides measuring $6 - 3 = 3$ cm.
Step 3: The area of the new square is calculated using the formula $\text{Area} = \text{side}^2$.
Thus, the new area is $3^2 = 9 \ \text{cm}^2$.

Therefore, the solution to the problem is $9 \ \text{cm}^2$.