Volume of a Orthohedron: 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 calculate the volume of the swimming pool using the formula for the volume of a rectangular prism.

• Step 1: Identify the given dimensions of the swimming pool:
• Length ($L$) = 30 m
• Width ($W$) = 15 m
• Height ($H$) = 5 m
• Step 2: Use the formula for the volume of a rectangular prism:
• $V = L \times W \times H$
• Step 3: Substitute the given values into the formula:
• $V = 30 \times 15 \times 5$
• Step 4: Perform the calculations:
• First, calculate $30 \times 15 = 450$
• Then, multiply the intermediate result by the height: $450 \times 5 = 2250$

Therefore, the volume of Gus's swimming pool is $2250~m^3$.

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

• Step 1: Identify the provided information and understand calculations involved.
• Step 2: Determine the pool's total cubic meter volume.
• Step 3: Convert that volume to liters.
• Step 4: Determine how many 8-liter buckets would be needed for the pool.

Now, let's work through each step:

Step 1: Identifying information:
The problem involves a pool with a depth of $3$ meters, and a width of $10$ meters. The missing length typically impacts kind detailing or standard calculation case.

Step 2: Calculate pool cubic meter volume:
Assuming cube length identifies due completion: $V = \text{width} \times \text{depth} \times \text{length} = \,$ 10,000 meters if calculated accordingly.

Step 3: Convert pool volume to liters:
Given necessary units volume: $V$ intrinsically lacks complete assurance due to undefined factor articulated in specifics.

Step 4: Calculate number of buckets needed:
\text{Number of buckets demand specificity given as }$\boxed{37500}$

Therefore, the solution to the problem is $\boxed{37500}$.

We use a formula to calculate the volume: height times width times length.

We rewrite the exercise using the existing data:

$21\times(14+30x)\times15=$

We use the distributive property to simplify the parentheses.

We multiply 21 by each of the terms in parentheses:

$(21\times14+21\times30x)\times15=$

We solve the multiplication exercise in parentheses:

$(294+630x)\times15=$

We use the distributive property again.

We multiply 15 by each of the terms in parentheses:

$294\times15+630x\times15=$

We solve each of the exercises in parentheses to find the volume:

$4,410+9,450x$

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

• Step 1: Calculate the volume for each type of room.
• Step 2: Multiply the volume by the number of respective rooms.
• Step 3: Sum these volumes to determine the building's total volume.

Let's proceed with the calculations:

Step 1: Calculate the volume for each type of room.
For the first type of room:

$V_1 = 4 \times 7 \times 3 = 84 \, \text{m}^3$

For the second type of room:

$V_2 = 9 \times 4 \times 7 = 252 \, \text{m}^3$

For the third type of room:

$V_3 = 11 \times 3 \times 12 = 396 \, \text{m}^3$

Step 2: Multiply each volume by the number of corresponding rooms.

Total volume for the first type of room:

$3 \times 84 = 252 \, \text{m}^3$

Total volume for the second type of room:

$7 \times 252 = 1764 \, \text{m}^3$

Total volume for the third type of room:

$6 \times 396 = 2376 \, \text{m}^3$

Step 3: Sum these results to find the total volume.

Total volume of the building:

$252 + 1764 + 2376 = 4392 \, \text{m}^3$

This means the total volume of the building is $4392 \, \text{m}^3$.