The volume of a rectangular prism is $(x^3-3 x^2+5 x-3)$, and the area of its base is $(x^2-2)$. If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?
A. $x-3+\frac{7 x-9}{x^2-2}$
B. $x-3+\frac{7 x-9}{x^3-3 x^2+5 x-3}$
C. $x-3+\frac{7 x+3}{x^2-2}$
D. $x-3+\frac{7 x+3}{x^3-3 x^2+5 x-3}$
Answer (2)
Divide the volume of the rectangular prism by the area of its base to find the height.
Perform polynomial long division of x 3 − 3 x 2 + 5 x − 3 by x 2 − 2 .
The result of the division is x − 3 with a remainder of 7 x − 9 .
Express the height as H = x − 3 + x 2 − 2 7 x − 9 .
Explanation
Understanding the Problem The problem states that the volume of a rectangular prism is given by V = x 3 − 3 x 2 + 5 x − 3 , and the area of its base is B = x 2 − 2 . We are also told that the volume of a rectangular prism is the product of its base area and height, i.e., V = B × H , where H is the height. Our objective is to find the height H of the prism.
Finding the Height Since V = B × H , we can find the height by dividing the volume by the base area: H = B V = x 2 − 2 x 3 − 3 x 2 + 5 x − 3 . To find the height, we need to perform polynomial long division.
Polynomial Long Division Performing polynomial long division of ( x 3 − 3 x 2 + 5 x − 3 ) by ( x 2 − 2 ) , we get:
x - 3
____________________
x^2-2 | x^3 - 3x^2 + 5x - 3
-(x^3 - 2x)
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- 3x^2 + 7x - 3
-(- 3x^2 + 6)
____________________
7x - 9
So, the quotient is x − 3 and the remainder is 7 x − 9 . Therefore, the height H can be expressed as:
H = x − 3 + x 2 − 2 7 x − 9
Final Answer The height of the prism is x − 3 + x 2 − 2 7 x − 9 .
Examples
Understanding polynomial division can help in various real-world scenarios, such as determining the dimensions of a garden. For instance, if you know the area of the garden and one of its dimensions can be expressed as a polynomial, you can use polynomial division to find the expression for the other dimension. This is also applicable in engineering when calculating volumes and dimensions of structures with complex shapes.
The height of the rectangular prism is given by the formula H = x − 3 + x 2 − 2 7 x − 9 . This result is found by dividing the volume of the prism by the area of its base using polynomial long division. After performing the division, we arrive at this expression for the height.
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