The volume of a right rectangular solid is 192 ft3. Find the height of the solid, if the length is 2 times the width, and the height is twice the perimeter of the base.

To solve this problem, let's denote the dimensions of the right rectangular solid as follows:

- Let \( w \) be the width of the solid (in feet).
- Then, the length \( l \) is \( 2w \) (since the length is 2 times the width).
- The height \( h \) is given as twice the perimeter of the base.

### Step 1: Calculate the perimeter of the base
The base of the solid is a rectangle with length \( l = 2w \) and width \( w \). The perimeter \( P \) of this rectangle is:

\[
P = 2(l + w) = 2(2w + w) = 2(3w) = 6w
\]

### Step 2: Express the height in terms of the width
The height \( h \) is twice the perimeter of the base:

\[
h = 2P = 2 \times 6w = 12w
\]

### Step 3: Use the volume formula
The volume \( V \) of a right rectangular solid is given by:

\[
V = l \times w \times h
\]

Substituting the expressions for \( l \) and \( h \):

\[
192 = (2w) \times w \times (12w)
\]

### Step 4: Simplify and solve for \( w \)
Simplifying the equation:

\[
192 = 24w^3
\]

To solve for \( w \), divide both sides by 24:

\[
w^3 = \frac{192}{24} = 8
\]

Now, take the cube root of both sides:

\[
w = \sqrt[3]{8} = 2
\]

### Step 5: Find the height \( h \)
Now that \( w = 2 \) feet, substitute this into the expression for height:

\[
h = 12w = 12 \times 2 = 24 \text{ feet}
\]

### Final Answer
The height of the solid is \( 24 \) feet.

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How would the problem change if the volume were different?
2. What would happen if the height was equal to the perimeter instead of twice the perimeter?
3. How can you generalize this method to other geometric shapes?
4. What is the relationship between surface area and volume in this solid?
5. How would the dimensions change if the height were fixed and the volume were to increase?

**Tip:** When working with geometric problems involving volume, always start by writing down the known formulas and variables, and then methodically substitute the given values to solve for the unknowns.

The volume of a right rectangular solid is 192 ft^3. Find the height of the solid, if the length is 2 times the width, and the height is twice the perimeter of the base.

Solve the problem of finding the height of a rectangular solid with a volume of 192 ft^3, where the length is twice the width, and the height is twice the perimeter of the base. This problem involves geometric and algebraic concepts suitable for Grades 8-10.

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