If we increase the side of a cube by 6, how many times will the volume of the cube increase by?
If we increase the side of a cube by 6, how many times will the volume of the cube increase by?
A square has sides measuring 6 cm long. Another square has sides measuring 3 cm less than those of the previous square. Calculate the area of the new square.
Given a square whose sides are 5. We draw another square whose sides are longer by 5 than the length of the sides of the previous square. Find the area of the new square.
Given a square whose side length is 9.
A new square is formed with a side length that is three times smaller than the original.
Find the area of the new square.
Given a square whose side length is 4. We draw a new square so that its side is 2 times bigger than the sides of the given square. Find the area of the new square.
If we increase the side of a cube by 6, how many times will the volume of the cube increase 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:
Therefore the volume of the initial cube (meaning before increasing its edge) is:
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:
In the second step we simplified the expression for the new cube's volume by using the power rule for multiplication in parentheses:
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:
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 -when we increased its edge by a factor of 6,
The correct answer is b.
A square has sides measuring 6 cm long. Another square has sides measuring 3 cm less than those of the previous square. Calculate the area of the new square.
To solve this problem, we'll follow these steps:
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 cm.
Step 3: The area of the new square is calculated using the formula .
Thus, the new area is .
Therefore, the solution to the problem is .
9
Given a square whose sides are 5. We draw another square whose sides are longer by 5 than the length of the sides of the previous square. Find the area of the new square.
To solve this problem, let's follow these steps:
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:
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:
Therefore, the area of the new square is 100 square units.
Thus, the correct answer is option 3: 100.
100
Given a square whose side length is 9.
A new square is formed with a side length that is three times smaller than the original.
Find the area of the new square.
9
Given a square whose side length is 4. We draw a new square so that its side is 2 times bigger than the sides of the given square. Find the area of the new square.
64