Cubes: Worded problems

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, we'll follow these steps:

• Step 1: Calculate the side length of the cube.
• Step 2: Find the base area of the cube.
• Step 3: Determine the height the water reaches based on its volume.

Now, let's work through each step:

Step 1: The volume of the cube is $125 \, \text{cm}^3$. Since the volume of a cube is given by $s^3$, we have:

$s^3 = 125$

Taking the cube root of both sides gives:

$s = \sqrt[3]{125} = 5 \, \text{cm}$

Step 2: The base area of the cube is $s^2$, which is:

$(5 \, \text{cm})^2 = 25 \, \text{cm}^2$

Step 3: The volume of water is $75 \, \text{cm}^3$, and we need to find the height $h$ it reaches in the cube:

$75 = 25 \times h$

Solving for $h$:

$h = \frac{75}{25} = 3 \, \text{cm}$

Therefore, the height to which the water will reach is $3 \, \text{cm}$.

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

• Step 1: Calculate the total volume of the spheres.
• Step 2: Determine the volume remaining in the cube after inserting the spheres.
• Step 3: Calculate the ratio between the total volume of the spheres and the remaining volume of the cube.

Now, let's work through each step:

Step 1: The volume of each sphere is given as 10 cm$^3$, and there are 5 spheres. Thus, the total volume of the spheres is:
$5 \times 10 = 50$ cm$^3$.

Step 2: The volume of the cube is given as 125 cm$^3$. After inserting the spheres, the remaining volume of the cube is:
$125 - 50 = 75$ cm$^3$.

Step 3: Calculate the ratio of the total volume of the spheres to the remaining volume of the cube:
$\frac{50 \text{ cm}^3}{75 \text{ cm}^3} = \frac{2}{3}$.

Therefore, the ratio between the total volume of the spheres and the volume remaining in the cube is $\frac{2}{3}$.

To solve this problem, follow these steps:

• Step 1: Calculate the side length of the cube.
  - The volume of the cube is $64 \, \text{cm}^3$. Using the formula for the volume of a cube, $s^3 = 64$.
  - Find $s$: $s = \sqrt[3]{64} = 4 \, \text{cm}$.
• Step 2: Determine the base area of the cube.
  - The base of the cube is a square with side length $4 \, \text{cm}$.
  - The area, $A_{\text{base}}$, is $4 \, \text{cm} \times 4 \, \text{cm} = 16 \, \text{cm}^2$.
• Step 3: Calculate the height the sand will reach.
  - We have $32 \, \text{cm}^3$ of sand. Use the volume formula for height: $32 = 16 \times h$.
  - Solve for $h$: $h = \frac{32}{16} = 2 \, \text{cm}$.

Therefore, the sand will reach a height of $2 \, \text{cm}$ in the cube.

To solve this problem, we need to determine how many smaller cubes with a volume of 8 cm³ can fit inside a larger cube with a volume of 84 cm³.

We use the formula:

• Number of smaller cubes $n = \frac{\text{Volume of the large cube}}{\text{Volume of the smaller cube}}$.

Substituting the given volumes:

$n = \frac{84 \text{ cm}^3}{8 \text{ cm}^3} = 10.5$.

Since we can only fit entire cubes, we round down to the nearest whole number. Therefore, the number of entire 8 cm³ cubes that can fit is 10.

Therefore, the solution to the problem is $10$.