Find The Height Of A Rectangular Prism

by Alex Johnson 39 views

Hey there, math enthusiasts! Ever found yourself staring at the volume and base area of a rectangular prism and wondering how to figure out its height? Well, you've come to the right place! Today, we're diving deep into a problem that involves a bit of polynomial division, but don't worry, we'll break it down step-by-step so it’s super clear. Our mission, should we choose to accept it, is to find the expression that represents the height of a rectangular prism when we're given its volume and the area of its base. It's like a mathematical detective story, and the height is our prime suspect!

Unpacking the Problem: Volume, Base Area, and Height

Let's start by getting cozy with the core concepts. In the world of geometry, a rectangular prism is a 3D shape with six rectangular faces – think of a brick or a shoebox. The volume of any prism is essentially the amount of space it occupies. For a rectangular prism, this is calculated by multiplying the area of its base by its height. So, we have a fundamental formula: Volume = Base Area × Height. This relationship is the lynchpin of our entire problem. If we know the volume and the base area, we can rearrange this formula to solve for the height: Height = Volume / Base Area. See? It's already starting to look solvable!

Now, let's look at the specific expressions we've been given. The problem states that the volume of our rectangular prism is represented by the polynomial ${10 x^3+46 x^2-21 x-27}$. This is a cubic polynomial, meaning the highest power of 'x' is 3. On the other hand, the area of the base is given by the expression ${2 x^2+8 x-9}$, which is a quadratic polynomial (the highest power of 'x' is 2). Our goal is to find the expression for the height, which, according to our formula, will be the result of dividing the volume polynomial by the base area polynomial. This means we'll be performing polynomial long division. It might sound a little daunting, but trust me, it's just an organized way of doing division with polynomials, similar to how you learned to divide numbers.

We're looking for an expression that, when multiplied by ${2 x^2+8 x-9}$ (the base area), gives us ${10 x^3+46 x^2-21 x-27}$ (the volume). This is precisely what polynomial division achieves. The result of this division will be our mysterious height expression. So, get ready to roll up your sleeves and dive into the fascinating world of polynomial division. We'll tackle it piece by piece, ensuring that by the end, you'll feel confident in your ability to solve similar problems. Remember, every complex mathematical problem is just a series of simpler steps waiting to be discovered!

The Tool of the Trade: Polynomial Long Division

Alright, let's get down to business with polynomial long division. This is our primary tool for solving this problem. Think of it as the granddaddy of algebraic division, and it works much like the long division you learned in elementary school, but with variables and exponents. We're going to divide the volume polynomial, ${10 x^3+46 x^2-21 x-27}$, by the base area polynomial, ${2 x^2+8 x-9}$. The result of this division will be the expression for the height.

To set up the division, we write it in a format similar to numerical long division: the dividend (volume) goes inside the division bracket, and the divisor (base area) goes outside to the left. It should look something like this:

        ____________
2x^2+8x-9 | 10x^3 + 46x^2 - 21x - 27

Our strategy is to take it one term at a time. We look at the leading term of the dividend ($10x^3}$) and the leading term of the divisor (${2x^2}$). We ask ourselves "What do I need to multiply ${2x^2$ by to get ${10x^3}$?" The answer is ${5x}$ (because ${2x^2 \times 5x = 10x^3}$). This ${5x}$ is the first term of our quotient (which will be our height expression).

We then write $5x}$ above the ${x^2}$ term in the dividend. After that, we multiply the entire divisor (${2x^2+8x-9}$) by this term (${5x}$) and write the result below the dividend, aligning like terms ${5x \times (2x^2+8x-9) = 10x^3 + 40x^2 - 45x$.

        5x _________
2x^2+8x-9 | 10x^3 + 46x^2 - 21x - 27
          -(10x^3 + 40x^2 - 45x)
          ---------------------

Next, we subtract this result from the corresponding terms in the dividend. It's crucial to remember to distribute the negative sign to each term being subtracted. So, ${(10x^3 + 46x^2 - 21x) - (10x^3 + 40x^2 - 45x)}$ becomes ${10x^3 - 10x^3 + 46x^2 - 40x^2 - 21x - (-45x)}$, which simplifies to ${6x^2 + 24x}$. We then bring down the next term from the dividend (the -27).

        5x _________
2x^2+8x-9 | 10x^3 + 46x^2 - 21x - 27
          -(10x^3 + 40x^2 - 45x)
          ---------------------
                 6x^2 + 24x - 27

Now, we repeat the process. We look at the leading term of our new polynomial ($6x^2}$) and the leading term of the divisor (${2x^2}$). We ask "What do I need to multiply ${2x^2$ by to get ${6x^2}$?" The answer is $+3$. This $+3$ is the next term in our quotient.

We multiply the entire divisor ($2x^2+8x-9}$) by $+3$ ${3 \times (2x^2+8x-9) = 6x^2 + 24x - 27$. We write this result below our current polynomial and subtract.

        5x + 3
2x^2+8x-9 | 10x^3 + 46x^2 - 21x - 27
          -(10x^3 + 40x^2 - 45x)
          ---------------------
                 6x^2 + 24x - 27
               -(6x^2 + 24x - 27)
               -----------------
                      0

Since our remainder is 0, the division is exact. This means that ${5x + 3}$ is the expression that perfectly represents the height of our rectangular prism. We've successfully used polynomial long division to unravel the mystery!

The Grand Finale: The Height Expression

So, after performing the meticulous process of polynomial long division, we've arrived at our answer. The division of the volume polynomial ${10 x^3+46 x^2-21 x-27}$ by the base area polynomial ${2 x^2+8 x-9}$ yielded a quotient of ${5x + 3}$ and a remainder of 0. This is fantastic news! It means that the base area multiplied by ${5x + 3}$ exactly equals the volume.

Therefore, the expression that represents the height of the rectangular prism is $5x + 3$. This is our final answer, the solution to our mathematical puzzle. It's a simple linear expression, which makes sense – height is often a linear dimension. We found it by trusting in the fundamental relationship between volume, base area, and height, and by employing the powerful tool of polynomial long division.

This process underscores a crucial concept in algebra and geometry: the connection between symbolic representations and geometric properties. Polynomials aren't just abstract strings of symbols; they can describe real-world quantities like dimensions, areas, and volumes. By mastering techniques like polynomial division, we gain the ability to extract meaningful information and solve problems that combine these different mathematical domains.

Think about it – if we knew a specific value for 'x', we could plug it into ${5x + 3}$ to find the actual height of the prism. For example, if x = 2, the height would be ${5(2) + 3 = 13}$. Similarly, we could calculate the base area and volume for x = 2 and verify that ${Base Area \times Height = Volume}$. This demonstrates the practical application of algebraic expressions in understanding geometric shapes.

We successfully navigated the complexities of dividing polynomials, breaking down a seemingly intricate problem into manageable steps. The key was understanding the relationship Height = Volume / Base Area and systematically applying the long division algorithm. Each step involved finding a term that would eliminate the leading term of the current polynomial, much like chipping away at a block of marble to reveal a sculpture within.

This problem is a perfect example of how algebra provides the language and tools to describe and manipulate geometric concepts. The fact that the division resulted in a zero remainder confirms that ${2 x^2+8 x-9}$ is indeed a factor of ${10 x^3+46 x^2-21 x-27}$, and ${5x + 3}$ is the corresponding co-factor representing the height. It's a beautiful synergy between different branches of mathematics, showcasing the elegance and power of algebraic methods.

We’ve confirmed that our height expression is ${5x + 3}$. It’s a clean, straightforward result that perfectly fits the context of the problem. So, whenever you encounter a similar situation, remember the power of polynomial long division and the fundamental geometric formula. You've got this!

Verifying the Result

To be absolutely sure, let's quickly verify our result. We found that the height is ${5x + 3}$. If this is correct, then multiplying the base area ${(2 x^2+8 x-9)}$ by the height ${(5x + 3)}$ should give us back the original volume ${(10 x^3+46 x^2-21 x-27)}$. Let's do this multiplication:

(2x2+8x−9)×(5x+3)(2 x^2+8 x-9) \times (5x + 3)

We distribute each term in the first polynomial to each term in the second:

=2x2(5x+3)+8x(5x+3)−9(5x+3)= 2x^2(5x+3) + 8x(5x+3) - 9(5x+3)

Now, distribute further:

=(10x3+6x2)+(40x2+24x)−(45x+27)= (10x^3 + 6x^2) + (40x^2 + 24x) - (45x + 27)

Combine like terms:

=10x3+(6x2+40x2)+(24x−45x)−27= 10x^3 + (6x^2 + 40x^2) + (24x - 45x) - 27

=10x3+46x2−21x−27= 10x^3 + 46x^2 - 21x - 27

And there you have it! The result of our multiplication perfectly matches the given volume expression. This confirms that our calculated height of ${5x + 3}$ is indeed correct. It’s always a good practice to double-check your work, especially in mathematics, to ensure accuracy and build confidence in your solutions.

Conclusion

In conclusion, finding the height of a rectangular prism when given its volume and base area boils down to a fundamental geometric relationship and a powerful algebraic tool. By understanding that Volume = Base Area × Height, we can rearrange this to Height = Volume / Base Area. Applying polynomial long division to the given expressions, ${10 x^3+46 x^2-21 x-27}$ for the volume and ${2 x^2+8 x-9}$ for the base area, we systematically divided the former by the latter. This process led us to a quotient of ${5x + 3}$ and a remainder of 0. Therefore, the expression representing the height of the rectangular prism is $5x + 3$. We further verified this result through multiplication, confirming its accuracy. This problem beautifully illustrates how algebraic manipulation can unlock geometric insights, making abstract concepts tangible and solvable.

If you're interested in learning more about the properties of rectangular prisms and volumes, I recommend checking out resources like Khan Academy. They offer fantastic explanations and practice problems that can deepen your understanding of geometry and algebra.