the volume of a cuboid whith Dimensions 4, x, and (x + 3) is 216. write a formula for the volume of the cuboid. Using the formula created show that the volume of this cuboid is given by x + 3x - 54 = 0. Lastly write down the actual dimensions of the cuboid 

### Step 1: Write a formula for the volume of the cuboid

The volume \( V \) of a cuboid is given by the product of its length, width, and height. For a cuboid with dimensions 4, \( x \), and \( (x + 3) \), the volume formula is:

\[
V = 4 \times x \times (x + 3)
\]

### Step 2: Use the formula to find the equation given by \( x + 3x - 54 = 0 \)

Given that the volume of the cuboid is 216 cubic units, we substitute \( V = 216 \) into the formula:

\[
4 \times x \times (x + 3) = 216
\]

Simplify this equation:

\[
4x(x + 3) = 216
\]

Divide both sides by 4:

\[
x(x + 3) = 54
\]

Expanding the left side:

\[
x^2 + 3x = 54
\]

Rearrange to form a standard quadratic equation:

\[
x^2 + 3x - 54 = 0
\]

This matches the given equation.

### Step 3: Solve the quadratic equation to find the dimensions of the cuboid

Solve the quadratic equation \( x^2 + 3x - 54 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For this equation, \( a = 1 \), \( b = 3 \), and \( c = -54 \). Substitute these values:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-54)}}{2(1)}
\]

\[
x = \frac{-3 \pm \sqrt{9 + 216}}{2}
\]

\[
x = \frac{-3 \pm \sqrt{225}}{2}
\]

\[
x = \frac{-3 \pm 15}{2}
\]

So, we have two possible solutions:

\[
x = \frac{-3 + 15}{2} = 6 \quad \text{and} \quad x = \frac{-3 - 15}{2} = -9
\]

Since dimensions cannot be negative, we take \( x = 6 \).

### Step 4: Find the actual dimensions of the cuboid

Now, substitute \( x = 6 \) into the dimensions:

- Length = 4
- Width = \( x = 6 \)
- Height = \( x + 3 = 6 + 3 = 9 \)

Thus, the actual dimensions of the cuboid are:

\[
\text{Dimensions: } 4, 6, \text{ and } 9
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the volume change if \( x \) were a different value?
2. What are the properties of quadratic equations that help solve for dimensions?
3. How can we verify that the calculated dimensions yield the correct volume?
4. Can you solve similar problems with other shapes like cylinders or spheres?
5. How does the discriminant help determine the nature of solutions in quadratic equations?

**Tip:** Always check if your solution makes sense in the context of the problem, especially when dealing with physical dimensions.

The volume of a cuboid with dimensions 4, x, and (x + 3) is 216. Write a formula for the volume of the cuboid. Using the formula created, show that the volume of this cuboid is given by x^2 + 3x - 54 = 0. Lastly, write down the actual dimensions of the cuboid.

Learn how to calculate the volume of a cuboid with dimensions 4, x, and (x + 3) using algebra and quadratic equations. This detailed solution shows how to derive the equation x^2 + 3x - 54 = 0 and solve for the cuboid's dimensions. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation