Box Volume: Find The Height Expression
Hey guys! Let's dive into a cool problem involving the volume of a rectangular prism, also known as a box. We're given the volume function, the length, and the width, and our mission is to find the expression that represents the height. Sounds like a fun math adventure, right? Let's break it down step by step so we can all understand it clearly.
Understanding the Volume of a Rectangular Prism
Okay, first things first, let's refresh our memory about the volume of a rectangular prism. The volume, often denoted as V, is the amount of space inside the box. Imagine filling the box with water; the amount of water you can pour in is the volume. The formula for the volume of a rectangular prism is super straightforward:
V = length × width × height
In mathematical terms, we often use letters to represent these dimensions. So, if we let l be the length, w be the width, and h be the height, the formula becomes:
V = l × w × h
This formula is the key to solving our problem. We know the volume V as a function f(x), and we know the expressions for the length and width. Our goal is to find the expression for the height. To find the height, we'll use a bit of algebraic manipulation. Think of it like solving a puzzle – we have all the pieces, we just need to put them together in the right way.
Now, in our specific problem, we're given the volume as a cubic function, which might sound intimidating, but don't worry, we'll tackle it together. We have f(x) = x³ - x² - 17x - 15 as the volume. We also know that the length is (x - 5) and the width is (x + 1). We need to find the height, which will also be an expression in terms of x. So, the equation we're working with looks like this:
x³ - x² - 17x - 15 = (x - 5) × (x + 1) × h
Our main task now is to figure out what that h is. We'll do this by using our algebra skills to rearrange the equation and isolate h. This is where the fun begins – we're going to use polynomial division or factoring to solve for the height. Stay with me, and we'll get through this together!
Setting Up the Problem
Let's recap what we know so far. We have the volume of the box represented by the function f(x) = x³ - x² - 17x - 15. We also know the length is (x - 5) and the width is (x + 1). Remember, the volume of a rectangular prism is calculated by multiplying the length, width, and height. So, we can write the equation:
x³ - x² - 17x - 15 = (x - 5)(x + 1)h
where h represents the height, which is what we're trying to find. To find the expression for the height, we need to isolate h on one side of the equation. This means we need to divide the volume function by the product of the length and width. In other words, we'll divide x³ - x² - 17x - 15 by (x - 5)(x + 1).
Before we jump into the division, let’s multiply the length and width expressions together. This will simplify our division process. So, let's multiply (x - 5) and (x + 1):
(x - 5)(x + 1) = x(x + 1) - 5(x + 1) = x² + x - 5x - 5 = x² - 4x - 5
Now we have the product of the length and width, which is x² - 4x - 5. Our equation now looks like this:
x³ - x² - 17x - 15 = (x² - 4x - 5)h
To find h, we'll perform polynomial division. We'll divide the volume x³ - x² - 17x - 15 by the product of the length and width x² - 4x - 5. Polynomial division might seem a bit tricky at first, but with a little practice, it becomes much easier. Think of it like long division with numbers, but now we're working with polynomials. We’ll set up the division and go through the steps carefully to find the height h.
Performing Polynomial Division
Alright, let's get into the nitty-gritty of polynomial division. We're going to divide the volume, x³ - x² - 17x - 15, by the product of the length and width, which we found to be x² - 4x - 5. Set up the long division like this:
__________
x² - 4x - 5 | x³ - x² - 17x - 15
First, we ask ourselves: what do we need to multiply x² by to get x³? The answer is x. So, we write x above the line:
x ______
x² - 4x - 5 | x³ - x² - 17x - 15
Now, multiply the entire divisor (x² - 4x - 5) by x:
x(x² - 4x - 5) = x³ - 4x² - 5x
Write this result below the dividend and subtract:
x ______
x² - 4x - 5 | x³ - x² - 17x - 15
-(x³ - 4x² - 5x)
------------------
3x² - 12x - 15
Bring down the next term, which is -15. Now we have 3x² - 12x - 15. Next, we ask: what do we need to multiply x² by to get 3x²? The answer is 3. So, we write +3 next to the x above the line:
x + 3
x² - 4x - 5 | x³ - x² - 17x - 15
-(x³ - 4x² - 5x)
------------------
3x² - 12x - 15
Multiply the divisor (x² - 4x - 5) by 3:
3(x² - 4x - 5) = 3x² - 12x - 15
Write this below and subtract:
x + 3
x² - 4x - 5 | x³ - x² - 17x - 15
-(x³ - 4x² - 5x)
------------------
3x² - 12x - 15
-(3x² - 12x - 15)
------------------
0
We get a remainder of 0, which means the division is exact. The quotient, x + 3, is the expression for the height.
Finding the Height Expression
So, after performing the polynomial division, we've found that the height, h, is represented by the expression (x + 3). This means that if you multiply the length (x - 5), the width (x + 1), and the height (x + 3), you will indeed get the volume x³ - x² - 17x - 15. Isn't it cool how it all fits together?
Let’s quickly verify our answer. We found that the height is (x + 3). So, we should have:
(x - 5)(x + 1)(x + 3) = x³ - x² - 17x - 15
We already know that (x - 5)(x + 1) = x² - 4x - 5, so now we multiply this by (x + 3):
(x² - 4x - 5)(x + 3) = x²(x + 3) - 4x(x + 3) - 5(x + 3)
= x³ + 3x² - 4x² - 12x - 5x - 15
= x³ - x² - 17x - 15
This matches the given volume function f(x), so we know our height expression is correct!
Therefore, the expression representing the height of the box is (x + 3). High fives all around – we nailed it!
Conclusion
Awesome job, guys! We successfully found the expression for the height of the rectangular prism. By understanding the formula for the volume of a rectangular prism and applying polynomial division, we were able to solve this problem step by step. Remember, the key is to break down the problem into smaller, manageable steps and take it one thing at a time.
We started by recalling the volume formula, V = l × w × h, and setting up the equation with the given volume function, length, and width. Then, we multiplied the length and width expressions together to simplify the division process. We performed polynomial division carefully, making sure to align the terms and subtract correctly. Finally, we verified our answer by multiplying the length, width, and height expressions together and confirming that they equal the given volume function.
Math problems like these can seem daunting at first, but with a solid understanding of the basics and a systematic approach, we can tackle them with confidence. Keep practicing, keep asking questions, and most importantly, keep enjoying the journey of learning! You've got this!
