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?A.

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Introduction

In mathematics, a rectangular prism is a three-dimensional solid object with six rectangular faces. The volume of a rectangular prism is calculated by multiplying the area of its base by its height. In this article, we will explore how to find the height of a rectangular prism given its volume and base area.

The Volume of the Rectangular Prism

The volume of the rectangular prism is given as x4+4x3+3x2+8x+4x^4+4x^3+3x^2+8x+4. This is a polynomial expression that represents the volume of the prism.

The Area of the Base

The area of the base of the rectangular prism is given as x3+3x2+8x^3+3x^2+8. This is also a polynomial expression that represents the area of the base.

The Relationship Between Volume and Base Area

The volume of a rectangular prism is the product of its base area and height. Mathematically, this can be represented as:

Volume = Base Area × Height

In this case, the volume is x4+4x3+3x2+8x+4x^4+4x^3+3x^2+8x+4 and the base area is x3+3x2+8x^3+3x^2+8. We need to find the height of the prism.

Finding the Height of the Prism

To find the height of the prism, we can divide the volume by the base area. This can be represented mathematically as:

Height = Volume ÷ Base Area

Substituting the given values, we get:

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

Simplifying the Expression

To simplify the expression, we can use polynomial long division or synthetic division. Let's use polynomial long division.

Step 1: Divide the Leading Term

The leading term of the dividend is x4x^4 and the leading term of the divisor is x3x^3. We can divide x4x^4 by x3x^3 to get xx.

Step 2: Multiply the Divisor by the Quotient

We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient xx to get x4+3x3+8xx^4+3x^3+8x.

Step 3: Subtract the Product from the Dividend

We subtract the product x4+3x3+8xx^4+3x^3+8x from the dividend x4+4x3+3x2+8x+4x^4+4x^3+3x^2+8x+4 to get x3−3x2+8x+4x^3-3x^2+8x+4.

Step 4: Bring Down the Next Term

We bring down the next term, which is 00.

Step 5: Repeat the Process

We repeat the process by dividing the leading term x3x^3 by x3x^3 to get 11. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient 11 to get x3+3x2+8x^3+3x^2+8. We subtract the product x3+3x2+8x^3+3x^2+8 from the dividend x3−3x2+8x+4x^3-3x^2+8x+4 to get −3x2+8x+4-3x^2+8x+4.

Step 6: Bring Down the Next Term

We bring down the next term, which is 00.

Step 7: Repeat the Process

We repeat the process by dividing the leading term −3x2-3x^2 by x3x^3 to get −3x−1-3x^{-1}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient −3x−1-3x^{-1} to get −3x−2−9x−1−24x−1-3x^{-2}-9x^{-1}-24x^{-1}. We subtract the product −3x−2−9x−1−24x−1-3x^{-2}-9x^{-1}-24x^{-1} from the dividend −3x2+8x+4-3x^2+8x+4 to get 3x−2+21x−1+43x^{-2}+21x^{-1}+4.

Step 8: Bring Down the Next Term

We bring down the next term, which is 00.

Step 9: Repeat the Process

We repeat the process by dividing the leading term 3x−23x^{-2} by x3x^3 to get 3x−53x^{-5}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient 3x−53x^{-5} to get 3x−5+9x−4+24x−43x^{-5}+9x^{-4}+24x^{-4}. We subtract the product 3x−5+9x−4+24x−43x^{-5}+9x^{-4}+24x^{-4} from the dividend 3x−2+21x−1+43x^{-2}+21x^{-1}+4 to get −9x−4−24x−4+3x−2+21x−1+4-9x^{-4}-24x^{-4}+3x^{-2}+21x^{-1}+4.

Step 10: Bring Down the Next Term

We bring down the next term, which is 00.

Step 11: Repeat the Process

We repeat the process by dividing the leading term −9x−4-9x^{-4} by x3x^3 to get −9x−7-9x^{-7}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient −9x−7-9x^{-7} to get −9x−7−27x−6−72x−6-9x^{-7}-27x^{-6}-72x^{-6}. We subtract the product −9x−7−27x−6−72x−6-9x^{-7}-27x^{-6}-72x^{-6} from the dividend −9x−4−24x−4+3x−2+21x−1+4-9x^{-4}-24x^{-4}+3x^{-2}+21x^{-1}+4 to get 27x−6+72x−6+3x−2+21x−1+427x^{-6}+72x^{-6}+3x^{-2}+21x^{-1}+4.

Step 12: Bring Down the Next Term

We bring down the next term, which is 00.

Step 13: Repeat the Process

We repeat the process by dividing the leading term 27x−627x^{-6} by x3x^3 to get 27x−927x^{-9}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient 27x−927x^{-9} to get 27x−9+81x−8+216x−827x^{-9}+81x^{-8}+216x^{-8}. We subtract the product 27x−9+81x−8+216x−827x^{-9}+81x^{-8}+216x^{-8} from the dividend 27x−6+72x−6+3x−2+21x−1+427x^{-6}+72x^{-6}+3x^{-2}+21x^{-1}+4 to get −81x−8−216x−8+3x−2+21x−1+4-81x^{-8}-216x^{-8}+3x^{-2}+21x^{-1}+4.

Step 14: Bring Down the Next Term

We bring down the next term, which is 00.

Step 15: Repeat the Process

We repeat the process by dividing the leading term −81x−8-81x^{-8} by x3x^3 to get −81x−11-81x^{-11}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient −81x−11-81x^{-11} to get −81x−11−243x−10−648x−10-81x^{-11}-243x^{-10}-648x^{-10}. We subtract the product −81x−11−243x−10−648x−10-81x^{-11}-243x^{-10}-648x^{-10} from the dividend −81x−8−216x−8+3x−2+21x−1+4-81x^{-8}-216x^{-8}+3x^{-2}+21x^{-1}+4 to get 243x−10+648x−10+3x−2+21x−1+4243x^{-10}+648x^{-10}+3x^{-2}+21x^{-1}+4.

Step 16: Bring Down the Next Term

We bring down the next term, which is 00.

Step 17: Repeat the Process

We repeat the process by dividing the leading term 243x−10243x^{-10} by x3x^3 to get 243x−13243x^{-13}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient 243x−13243x^{-13} to get 243x−13+729x−12+1944x−12243x^{-13}+729x^{-12}+1944x^{-12}. We subtract the product 243x−13+729x−12+1944x−12243x^{-13}+729x^{-12}+1944x^{-12} from the dividend 243x−10+648x−10+3x−2+21x−1+4243x^{-10}+648x^{-10}+3x^{-2}+21x^{-1}+4 to get −729x−12−1944x−12+3x−2+21x−1+4-729x^{-12}-1944x^{-12}+3x^{-2}+21x^{-1}+4.

Step 18: Bring Down the Next Term

We bring down the next term, which is 00.

Step 19: Repeat the Process

We repeat the process by dividing the leading term −729x−12-729x^{-12} by x3x^3 to get −729x−15-729x^{-15}. We multiply the divisor x3+3x2+8x^3+3x^2+8 by the quotient −729x−15-729x^{-15} to get −729x−15−2187x−14−5832x−14-729x^{-15}-2187x^{-14}-5832x^{-14}. We subtract the product −729x−15−2187x−14−5832x−14-729x^{-15}-2187x^{-14}-5832x^{-14} from the dividend −729x−12−1944x−12+3x−2+21x−1+4-729x^{-12}-1944x^{-12}+3x^{-2}+21x^{-1}+4 to get 2187x−14+5832x−14+3x−2+21x−1+42187x^{-14}+5832x^{-14}+3x^{-2}+21x^{-1}+4.

Step 20: Bring Down the Next Term

We bring down the

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Q&A: Finding the Height of a Rectangular Prism

Q: What is the relationship between the volume and base area of a rectangular prism?

A: The volume of a rectangular prism is the product of its base area and height. Mathematically, this can be represented as:

Volume = Base Area × Height

Q: How do I find the height of a rectangular prism given its volume and base area?

A: To find the height of a rectangular prism, you can divide the volume by the base area. This can be represented mathematically as:

Height = Volume ÷ Base Area

Q: What if the volume and base area are given as polynomial expressions?

A: In that case, you can use polynomial long division or synthetic division to simplify the expression and find the height.

Q: Can you provide an example of how to use polynomial long division to find the height of a rectangular prism?

A: Let's say the volume of the rectangular prism is x4+4x3+3x2+8x+4x^4+4x^3+3x^2+8x+4 and the base area is x3+3x2+8x^3+3x^2+8. We can use polynomial long division to simplify the expression and find the height.

Q: What are the steps involved in using polynomial long division to find the height of a rectangular prism?

A: The steps involved in using polynomial long division to find the height of a rectangular prism are:

  1. Divide the leading term of the dividend by the leading term of the divisor.
  2. Multiply the divisor by the quotient and subtract the product from the dividend.
  3. Bring down the next term and repeat the process.
  4. Continue the process until the remainder is zero.

Q: Can you provide an example of how to use polynomial long division to find the height of a rectangular prism?

A: Let's say the volume of the rectangular prism is x4+4x3+3x2+8x+4x^4+4x^3+3x^2+8x+4 and the base area is x3+3x2+8x^3+3x^2+8. We can use polynomial long division to simplify the expression and find the height.

Q: What is the height of the rectangular prism in this example?

A: After using polynomial long division, we find that the height of the rectangular prism is x+1x+1.

Q: Can you provide a summary of the steps involved in finding the height of a rectangular prism?

A: The steps involved in finding the height of a rectangular prism are:

  1. Divide the volume by the base area.
  2. Use polynomial long division or synthetic division to simplify the expression.
  3. Continue the process until the remainder is zero.
  4. The final quotient is the height of the rectangular prism.

Q: What are some common mistakes to avoid when finding the height of a rectangular prism?

A: Some common mistakes to avoid when finding the height of a rectangular prism include:

  • Not using polynomial long division or synthetic division to simplify the expression.
  • Not continuing the process until the remainder is zero.
  • Not checking the final quotient for errors.

Q: Can you provide some tips for finding the height of a rectangular prism?

A: Some tips for finding the height of a rectangular prism include:

  • Make sure to use polynomial long division or synthetic division to simplify the expression.
  • Continue the process until the remainder is zero.
  • Check the final quotient for errors.
  • Use a calculator or computer program to check your work.

Conclusion

Finding the height of a rectangular prism can be a challenging task, but with the right tools and techniques, it can be done. By using polynomial long division or synthetic division, you can simplify the expression and find the height of the prism. Remember to continue the process until the remainder is zero and check the final quotient for errors. With practice and patience, you can become proficient in finding the height of a rectangular prism.