Math Problem Statement

The volume of a rectangular prism is (x^4 + 4x^3 + 3x^2 + 8x + 4), and the area of its base is (x^3 + 3x^2 + 8). If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?

Solution

The problem states that the volume of a rectangular prism is x4+4x3+3x2+8x+4x^4 + 4x^3 + 3x^2 + 8x + 4, and the area of the base is x3+3x2+8x^3 + 3x^2 + 8. To find the height of the prism, we use the fact that the volume is the product of the base area and the height, i.e.,

Volume=Base Area×Height\text{Volume} = \text{Base Area} \times \text{Height}

Thus, the height can be found by dividing the volume by the base area:

Height=VolumeBase Area=x4+4x3+3x2+8x+4x3+3x2+8\text{Height} = \frac{\text{Volume}}{\text{Base Area}} = \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8}

Now, let's perform this division:

Step 1: Polynomial long division

We need to divide x4+4x3+3x2+8x+4x^4 + 4x^3 + 3x^2 + 8x + 4 by x3+3x2+8x^3 + 3x^2 + 8.

  1. Divide the leading term x4x^4 by x3x^3, which gives xx.
  2. Multiply xx by x3+3x2+8x^3 + 3x^2 + 8, resulting in x4+3x3+8xx^4 + 3x^3 + 8x.
  3. Subtract this from the original polynomial:

(x4+4x3+3x2+8x+4)(x4+3x3+8x)=x3+3x2+4(x^4 + 4x^3 + 3x^2 + 8x + 4) - (x^4 + 3x^3 + 8x) = x^3 + 3x^2 + 4

Step 2: Repeat the division process

Now divide x3x^3 by x3x^3, which gives 11.

  1. Multiply 11 by x3+3x2+8x^3 + 3x^2 + 8, resulting in x3+3x2+8x^3 + 3x^2 + 8.
  2. Subtract:

(x3+3x2+4)(x3+3x2+8)=4(x^3 + 3x^2 + 4) - (x^3 + 3x^2 + 8) = -4

So the result of the division is:

x+14x3+3x2+8x + 1 - \frac{4}{x^3 + 3x^2 + 8}

This matches the third answer choice:

x+14x3+3x2+8x + 1 - \frac{4}{x^3 + 3x^2 + 8}

Final Answer:

The correct answer is the third option.

Would you like further details on any step?

Here are some related questions:

  1. How do you perform polynomial long division in general?
  2. What is the significance of dividing polynomials in finding the height of prisms?
  3. Can you apply this method to other types of solids?
  4. How would the problem change if the base area were a different polynomial?
  5. Can the height of a prism always be found this way?

Tip: When dividing polynomials, always align like terms to avoid mistakes in subtraction.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Volume of Rectangular Prisms
Algebra

Formulas

Volume = Base Area * Height
Height = Volume / Base Area

Theorems

Polynomial Long Division

Suitable Grade Level

Grades 10-12