Unveiling The Length: Rectangular Prism Volume Explained

by Alex Johnson 57 views

Hey math enthusiasts! Let's dive into a fun problem involving rectangular prisms, volume, and some algebraic manipulation. We're going to break down how to find the length of a box when we know its volume, width, and height. Get ready to flex those math muscles!

Decoding the Volume Function

Let's start with what we know. The volume of our rectangular prism (think of it as a fancy box) is given by the function x³ + 11x² + 20x - 32. This is a polynomial function, and it describes how the volume changes based on a variable, x. We are also told that the width of the box is x - 1, and the height is x + 8. Our mission? To figure out the expression that represents the length of the box.

To solve this, we'll use the fundamental formula for the volume of a rectangular prism:

Volume = Length × Width × Height

Since we know the volume, width, and height, we can rearrange this formula to solve for the length:

Length = Volume / (Width × Height)

This means we need to divide the volume function by the product of the width and height. Before we do that, let's make sure we have all the pieces ready. The volume is x³ + 11x² + 20x - 32, the width is x - 1, and the height is x + 8. Now we are ready to find the length of the box. Let's start this adventure by multiplying the width by the height.

Multiplying Width and Height

Before we can divide, we need to find the product of the width and height. This means multiplying (x - 1) by (x + 8).

Here's how we do it:

(x - 1) * (x + 8) = x * x + x * 8 - 1 * x - 1 * 8

This simplifies to:

x² + 8x - x - 8 = x² + 7x - 8

So, the product of the width and height is x² + 7x - 8. Now we know the area of the box. Next step, divide the volume by the area to find the length.

Finding the Length: The Division

Now, let's divide the volume function (x³ + 11x² + 20x - 32) by the result we just found for the area of the box, which is (x² + 7x - 8). This is where polynomial division comes in handy. It might seem daunting at first, but we'll break it down step by step to find the length of the box.

Here's how the division process looks:

  1. Divide the first term of the dividend by the first term of the divisor: x³ / x² = x This means we start with x as part of our quotient (the length).
  2. Multiply the divisor by the result: x * (x² + 7x - 8) = x³ + 7x² - 8x
  3. Subtract the result from the dividend: (x³ + 11x² + 20x - 32) - (x³ + 7x² - 8x) = 4x² + 28x - 32
  4. Bring down the next term (if any): In this case, all the terms were already used.
  5. Repeat the process: 4x² / x² = 4 So, we add +4 to our quotient.
  6. Multiply the divisor by the new result: 4 * (x² + 7x - 8) = 4x² + 28x - 32
  7. Subtract the result: (4x² + 28x - 32) - (4x² + 28x - 32) = 0

We end up with no remainder, which is a good sign! Our quotient is x + 4. Therefore, the expression representing the length of the box is x + 4.

Conclusion: The Final Answer

After all that algebraic work, we've found our answer! The expression representing the length of the box is x + 4. This corresponds to answer choice c. We have successfully used polynomial division and our understanding of rectangular prism volumes to solve the problem. High five!

This problem highlights how mathematical concepts like volume, polynomials, and division come together to solve real-world problems. It's not just about memorizing formulas; it's about understanding the relationships between different mathematical ideas and applying them logically. Keep practicing, and you'll become a master of these kinds of problems in no time. Congratulations, you are one step closer to mastering math!

I hope this helps you understand the problem. Please ask if you have any further questions. Keep learning and have fun!

For further reading and practice, you may want to visit these trusted websites:

  • Khan Academy: Offers comprehensive lessons and practice exercises on algebra, including polynomials and volume calculations.

  • Math is Fun: Provides clear explanations and examples of mathematical concepts, including geometry and algebra.