The volume of a rectangular prism is $(x^4+4 x^3+3 x^2+8 x+4)$, and the area of its base is $(x^3+3 x^2+8)$. If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?
A. $x+1-\frac{4}{x^4+4 x^3+3 x^2+8 x+4}$
B. $x+1+\frac{4}{x^4+4 x^3+3 x^2+8 x+4}$
C. $x+1-\frac{4}{x^3+3 x^2+8}$
D. $x+1+\frac{4}{x^3+3 x^2+8}$
Answer (1)
Divide the volume by the base area to find the height: h = A V .
Substitute the given expressions for V and A : h = x 3 + 3 x 2 + 8 x 4 + 4 x 3 + 3 x 2 + 8 x + 4 .
Perform polynomial long division to find the quotient and remainder.
Express the height as h = x + 1 − x 3 + 3 x 2 + 8 4 , so the final answer is x + 1 − x 3 + 3 x 2 + 8 4 .
Explanation
Finding the Height Formula We are given the volume V and the base area A of a rectangular prism, and we need to find the height h . We know that the volume of a rectangular prism is the product of its base area and height, so V = A h . Therefore, we can find the height by dividing the volume by the base area: $h =
\frac{V}{A}$.
Substituting Given Values We are given that the volume V = x 4 + 4 x 3 + 3 x 2 + 8 x + 4 and the base area A = x 3 + 3 x 2 + 8 . Substituting these expressions into the formula for the height, we get:
h = x 3 + 3 x 2 + 8 x 4 + 4 x 3 + 3 x 2 + 8 x + 4
Performing Polynomial Long Division To find the height, we need to perform polynomial long division. Dividing x 4 + 4 x 3 + 3 x 2 + 8 x + 4 by x 3 + 3 x 2 + 8 , we get a quotient of x + 1 and a remainder of − 4 . Therefore, we can write the height as:
h = x + 1 + x 3 + 3 x 2 + 8 − 4 = x + 1 − x 3 + 3 x 2 + 8 4
Final Answer Thus, the height of the prism is x + 1 − x 3 + 3 x 2 + 8 4 .
Examples
Understanding the volume and dimensions of rectangular prisms is crucial in various real-world applications. For instance, when designing a building, architects need to calculate the volume of rooms to determine heating and cooling requirements. Similarly, in packaging and logistics, knowing the volume of boxes and containers helps optimize storage and transportation. By relating the volume, base area, and height of a prism, we can solve practical problems involving space and capacity, ensuring efficient use of resources and accurate planning in construction, manufacturing, and logistics.
