Find The Height Of A Rectangular Prism

Hey guys! Let's dive into a cool math problem today that’s all about rectangular prisms. You know, those boxy shapes we see everywhere? We’ve got a problem here where we’re given the volume and the base area, and we need to figure out the height. It’s like a puzzle where we know the whole story and one of the pieces, and we need to find the missing part.

Understanding Rectangular Prisms: Volume, Base Area, and Height

Before we jump into solving this specific problem, let's quickly recap what we're dealing with. A rectangular prism is a 3D shape with six rectangular faces. Think of a brick, a shoebox, or even a swimming pool. The volume of a rectangular prism is the amount of space it occupies. It's calculated by multiplying its length, width, and height. However, there's a handy shortcut: volume = base area × height. The base area is simply the area of the bottom face (or any face, really, depending on how you orient it). So, if you multiply the area of the base by the height, you get the total volume. This relationship is super important because it’s the key to solving our problem today. We're given the volume and the base area as expressions involving 'x', and we need to find the height, which will also be an expression involving 'x'. This is a classic algebraic manipulation problem, and it's pretty straightforward once you get the hang of it. We'll be using polynomial division to find the height, which is essentially dividing the volume expression by the base area expression.

The Problem at Hand: Volume and Base Area Given

Alright, let’s look at the specifics of our problem. We are told that the volume of a rectangular prism is given by the expression $x^3 - 3x^2 + 5x - 3$. This is a cubic polynomial, which means the height and base area expressions, when multiplied together, result in this cubic form. We are also given that the area of its base is $x^2 - 2$. This is a quadratic polynomial. Our mission, should we choose to accept it, is to find the height of this prism. The problem explicitly states that the volume of a rectangular prism is the product of its base area and height. This means we can write the relationship as:

$$Volume = Base Area \times Height$$

To find the height, we just need to rearrange this formula:

$$Height = \frac{Volume}{Base Area}$$

So, in our case, we need to divide the polynomial $x^3 - 3x^2 + 5x - 3$ by the polynomial $x^2 - 2$. This is where polynomial long division comes into play. It's very similar to numerical long division, but with algebraic terms instead of numbers. We need to be careful with the order of terms and make sure we're subtracting correctly.

Solving for Height Using Polynomial Division

Now, let's get our hands dirty with the actual division. We want to divide $x^3 - 3x^2 + 5x - 3$ by $x^2 - 2$. It's important to write both polynomials in descending order of their powers. If any powers are missing, we can include them with a coefficient of zero as a placeholder, although in this case, our polynomials are already well-ordered. So, we set up the division like this:

        _____________
x^2 - 2 | x^3 - 3x^2 + 5x - 3

First, we look at the leading terms: $x^3$ (from the dividend) and $x^2$ (from the divisor). We ask ourselves, 'What do we need to multiply $x^2$ by to get $x^3$?' The answer is $x$. So, we write $x$ above the $5x$ term in the quotient area.

        x _________
x^2 - 2 | x^3 - 3x^2 + 5x - 3

Next, we multiply the entire divisor ($x^2 - 2$) by this term ($x$): $x(x^2 - 2) = x^3 - 2x$. We write this result below the dividend, aligning like terms.

        x _________
x^2 - 2 | x^3 - 3x^2 + 5x - 3
        -(x^3       - 2x)

Now, we subtract this from the dividend. Remember to change the signs of the terms being subtracted. So, $x^3 - x^3 = 0$, and $-3x^2 - 0 = -3x^2$. For the $x$ terms, we have $5x - (-2x) = 5x + 2x = 7x$. The constant term remains $-3$.

        x _________
x^2 - 2 | x^3 - 3x^2 + 5x - 3
        -(x^3       - 2x)
        ----------------
              -3x^2 + 7x - 3

Now, we bring down the next term from the dividend (which is already there: $-3x^2 + 7x - 3$). We repeat the process. We look at the new leading term, $-3x^2$, and the leading term of the divisor, $x^2$. 'What do we multiply $x^2$ by to get $-3x^2$?' The answer is $-3$. So, we write $-3$ in the quotient, next to the $x$.

        x - 3 _____
x^2 - 2 | x^3 - 3x^2 + 5x - 3
        -(x^3       - 2x)
        ----------------
              -3x^2 + 7x - 3

Now, we multiply the divisor ($x^2 - 2$) by this new term ($-3$): $-3(x^2 - 2) = -3x^2 + 6$. We write this below the current line, aligning like terms.

        x - 3 _____
x^2 - 2 | x^3 - 3x^2 + 5x - 3
        -(x^3       - 2x)
        ----------------
              -3x^2 + 7x - 3
              -(-3x^2     + 6)

Again, we subtract. $-3x^2 - (-3x^2) = -3x^2 + 3x^2 = 0$. Then, $7x - 0 = 7x$. And finally, $-3 - 6 = -9$.

        x - 3 _____
x^2 - 2 | x^3 - 3x^2 + 5x - 3
        -(x^3       - 2x)
        ----------------
              -3x^2 + 7x - 3
              -(-3x^2     + 6)
              ----------------
                     7x - 9

We stop here because the degree of the remainder ($7x - 9$, which is degree 1) is less than the degree of the divisor ($x^2 - 2$, which is degree 2). So, the result of our division is $x - 3$ with a remainder of $7x - 9$. This means:

$$\frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} = x - 3 + \frac{7x - 9}{x^2 - 2}$$

Wait a minute! The question asks for the height of the prism, and usually, when dealing with these kinds of problems in a typical math context, the division is expected to be exact, meaning there should be no remainder. Let me double-check the original problem statement and the provided options. Ah, I see the option provided is x - 3 + rac{7}{x^2-2}. It seems there might be a slight typo in the remainder part of the provided answer option, or perhaps in the original problem statement's coefficients. Let's assume for a moment that the problem intended for the division to be exact and the height to be a simple polynomial. If the height was simply $x-3$, then multiplying $(x^2-2)$ by $(x-3)$ would give $x^3 - 3x^2 - 2x + 6$. This is not our original volume. This indicates that the remainder is indeed part of the answer, and the question might be framed in a way that the height isn't a clean polynomial. However, let's re-examine the subtraction steps very carefully.

Let's re-do the subtraction:

Original problem: Volume = $x^3 - 3x^2 + 5x - 3$, Base Area = $x^2 - 2$.

        x - 3 _____
x^2 + 0x - 2 | x^3 - 3x^2 + 5x - 3

1. Divide $x^3$ by $x^2$ to get $x$. Multiply $x$ by $(x^2 - 2)$ to get $x^3 - 2x$. Subtract: $(x^3 - 3x^2 + 5x - 3) - (x^3 - 2x) = -3x^2 + 7x - 3$.

2. Divide $-3x^2$ by $x^2$ to get $-3$. Multiply $-3$ by $(x^2 - 2)$ to get $-3x^2 + 6$. Subtract: $(-3x^2 + 7x - 3) - (-3x^2 + 6) = 7x - 9$.

So the division result is indeed $x-3$ with a remainder of $7x-9$. The expression for the height is x-3 + rac{7x-9}{x^2-2}.

Now, let's look at the options provided in the question: A. x-3+rac{7}{x^2-2}. This option has a remainder of 7, not $7x-9$. This suggests a potential error in the problem statement or the given options. If the volume was $x^3 - 3x^2 + 3x - 6$, then dividing by $x^2-2$ would yield $x-3$ with remainder 0. If the volume was $x^3 - 3x^2 + 5x - 3$ and the base area was $x-1$, then height would be $x^2 - 2x + 3$.

Let's assume the provided answer option A, x-3+rac{7}{x^2-2}, is what the question is leading to, and try to work backward or see if there's a common mistake that leads to this. If the height were x-3 + rac{7}{x^2-2}, then the volume would be:

$(x^2 - 2) \times (x - 3 + \frac{7}{x^2-2}) = (x^2 - 2)(x - 3) + (x^2 - 2) \times \frac{7}{x^2-2}$

$= (x^3 - 3x^2 - 2x + 6) + 7$

$= x^3 - 3x^2 - 2x + 13$

This volume ($x^3 - 3x^2 - 2x + 13$) does not match the given volume ($x^3 - 3x^2 + 5x - 3$).

There appears to be an inconsistency in the problem statement or the provided options. However, based on standard polynomial division, the correct quotient is $x-3$ and the remainder is $7x-9$. If we strictly follow the calculation, the height is x-3 + rac{7x-9}{x^2-2}.

Given the option A is x-3+rac{7}{x^2-2}, and our calculation yields x-3+rac{7x-9}{x^2-2}, the closest match in structure is $x-3$ as the main part of the quotient. The remainder term is where the discrepancy lies. It's possible there's a typo in the original volume, base area, or the options.

Let's consider if the question meant that the height itself is a polynomial plus a rational term. In that case, our derived expression x-3 + rac{7x-9}{x^2-2} is mathematically correct based on the given volume and base area. The option A seems to have simplified the remainder incorrectly or had a different initial problem.

If we assume that the question intended the remainder to be just a constant '7', then working backwards, the division process would have to result in a remainder of 7. This would mean that after the subtraction of $-3x^2 + 6$, the remaining terms should have resulted in just 7. Let's see: $(-3x^2 + 7x - 3) - (-3x^2 + 6) = 7x - 9$. For this to be 7, the $7x$ term would have to disappear. This is not possible with the current numbers.

Conclusion based on strict calculation: The height is $\boldsymbol{x-3+\frac{7x-9}{x^2-2}}$.

Conclusion based on provided options and likely intent: Option A is $\boldsymbol{x-3+\frac{7}{x^2-2}}$. There is a discrepancy. If we are forced to choose the closest or intended answer from the options, and assuming there's a typo, the structure of $x-3$ as the polynomial part of the height is correct. The remainder term differs. However, without further clarification or correction, the mathematically derived answer is the most accurate. If this were a multiple-choice test and option A was the only one with $x-3$ as the polynomial part, it would be the most likely intended answer despite the remainder mismatch.

Let's re-examine the subtraction one last time to be absolutely sure.

We want to divide $V = x^3 - 3x^2 + 5x - 3$ by $B = x^2 - 2$.

$V = x(x^2 - 2) - 3x^2 + 5x - 3 + 2x$ $V = x(x^2 - 2) - 3x^2 + 7x - 3$

Now, consider the $-3x^2$ term. We can write $-3x^2$ as $-3(x^2 - 2) + 6$. So, $V = x(x^2 - 2) - 3(x^2 - 2) + 6 + 7x - 3$ $V = (x - 3)(x^2 - 2) + 7x + 3$

Ah, here is another small calculation error possibility. Let's be super careful with signs.

$V = x^3 - 3x^2 + 5x - 3$ $B = x^2 - 2$

1. $x^3 / x^2 = x$. $x(x^2 - 2) = x^3 - 2x$. Subtract from V: $(x^3 - 3x^2 + 5x - 3) - (x^3 - 2x) = -3x^2 + 7x - 3$.

2. $-3x^2 / x^2 = -3$. $-3(x^2 - 2) = -3x^2 + 6$. Subtract from the result of step 1: $(-3x^2 + 7x - 3) - (-3x^2 + 6) = -3x^2 + 7x - 3 + 3x^2 - 6 = 7x - 9$.

The remainder is definitely $7x - 9$. The quotient is $x-3$. So the height is $x-3 + \frac{7x-9}{x^2-2}$.

Given the provided answer option structure, it is highly probable that the question intended for the remainder to be a constant. If the remainder was intended to be 7, then the original volume might have been $x^3 - 3x^2 + 5x - 3 - (7x-9) + 7 = x^3 - 3x^2 - 2x + 13$. Or, if the divisor was different.

Let's consider another possibility: maybe the polynomial division was done incorrectly in the provided answer key, or the question itself has a typo. If we look at option A: x-3+rac{7}{x^2-2}. For this to be the height, then (x^2-2)(x-3+rac{7}{x^2-2}) must equal the volume. (x^2-2)(x-3) + (x^2-2)(rac{7}{x^2-2}) = (x^3 - 3x^2 - 2x + 6) + 7 = x^3 - 3x^2 - 2x + 13. This is not the given volume $x^3 - 3x^2 + 5x - 3$.

So, the provided option A is mathematically incorrect for the given volume and base area.

However, in a test scenario, you would typically select the answer that seems most plausible or follows the correct procedure partially. The polynomial part of the height is $x-3$. The remainder is $7x-9$. If option A were the only one starting with $x-3$ and having a rational term, it might be the intended answer due to a typo in the problem statement.

Final Answer Justification based on strict calculation: The height of the prism is the volume divided by the base area. Performing polynomial long division of $x^3 - 3x^2 + 5x - 3$ by $x^2 - 2$ yields a quotient of $x - 3$ and a remainder of $7x - 9$. Therefore, the height is $\boldsymbol{x-3+\frac{7x-9}{x^2-2}}$. Since this exact form is not given as an option, and option A shows a remainder of 7, there is likely an error in the question or the options provided. If forced to choose the option with the correct polynomial part, option A would be the closest choice, implying a typo in the remainder term. But mathematically, none of the options seem to be precisely correct given the problem statement.