The width of a rectangular box is 7 in  . The height is one fourth of the length x . The volume is 1372   .
(a) Write an equation in terms of x  that represents the given relationship.

(b) Solve the equation to find the dimensions of the given shape.

Let's go through the problem step by step.

### (a) Writing an Equation in Terms of \( x \):

Given:
- The width of the rectangular box is 7 inches.
- The height is one-fourth of the length \( x \), so height \( = \frac{x}{4} \).
- The volume of the box is 1372 cubic inches.

The volume \( V \) of a rectangular box is calculated using the formula:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

Substituting the known values:

\[
1372 = x \times 7 \times \frac{x}{4}
\]

This is the equation that represents the relationship in terms of \( x \).

### (b) Solving the Equation to Find the Dimensions:

Now, let's solve the equation:

\[
1372 = x \times 7 \times \frac{x}{4}
\]

Simplifying the right-hand side:

\[
1372 = \frac{7x^2}{4}
\]

Multiply both sides of the equation by 4 to eliminate the fraction:

\[
4 \times 1372 = 7x^2
\]

This simplifies to:

\[
5488 = 7x^2
\]

Now, divide both sides by 7:

\[
x^2 = \frac{5488}{7} = 784
\]

Taking the square root of both sides:

\[
x = \sqrt{784}
\]

The square root of 784 is:

\[
x = 28 \text{ inches}
\]

So, the length \( x \) is 28 inches.

Now that we have the length, we can find the height:

- Height \( = \frac{x}{4} = \frac{28}{4} = 7 \) inches

### Final Dimensions:
- **Length**: 28 inches
- **Width**: 7 inches
- **Height**: 7 inches

The box turns out to be a cube with all sides equal to 7 inches.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you solve equations with fractional coefficients?
2. What happens to the volume if the height is doubled but the width remains constant?
3. How does changing one dimension affect the overall volume of the box?
4. Can you derive the surface area for this cube-shaped box?
5. What are some real-world examples of objects with cubic shapes?

**Tip:** When solving quadratic equations, always check if the square root is a perfect square to simplify the process.

Learn how to solve a volume problem involving a rectangular box with specific dimensions using quadratic equations. Understand the relationship between length, width, height, and volume calculation. Suitable for Grades 7-9 students.

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