Given a prism with a right triangle base and the dimensions [tex]$h=x+1, b=x$[/tex], and [tex]$l=x+7$[/tex], what is a correct expression for the volume of the prism?
A. [tex]$V=\frac{1}{3}(x^3+8x^2+7x)$[/tex]
B. [tex]$V =x^2+8x+7$[/tex]
C. [tex]$V =x^3+8x^2+7x$[/tex]
D. [tex]$V=\frac{1}{2}(x^3+8x^2+7x)$[/tex]
Answer (1)
Calculate the area of the right triangle base: A = 2 1 × x × ( x + 1 ) .
Multiply the base area by the length of the prism to find the volume: V = A × l = 2 1 x ( x + 1 ) ( x + 7 ) .
Expand the expression: V = 2 1 ( x 3 + 8 x 2 + 7 x ) .
The correct expression for the volume is: V = 2 1 ( x 3 + 8 x 2 + 7 x ) .
Explanation
Analyze the problem and given data We are given a prism with a right triangle base. The dimensions are:
Height of the triangle: h = x + 1
Base of the triangle: b = x
Length of the prism: l = x + 7
We need to find the correct expression for the volume V of the prism.
Calculate the volume of the prism The volume of a prism is given by the formula: V = A × l where A is the area of the base and l is the length of the prism. Since the base is a right triangle, its area is given by: A = 2 1 × b × h Substituting the given values for b and h , we get: A = 2 1 × x × ( x + 1 ) A = 2 1 x ( x + 1 ) Now, substitute the expression for A and the given expression for l into the volume formula: V = 2 1 x ( x + 1 ) ( x + 7 ) Expand the expression to find the polynomial representation of the volume: V = 2 1 x ( x 2 + 7 x + x + 7 ) V = 2 1 x ( x 2 + 8 x + 7 ) V = 2 1 ( x 3 + 8 x 2 + 7 x ) So, the correct expression for the volume of the prism is: V = 2 1 ( x 3 + 8 x 2 + 7 x )
Identify the correct answer Comparing the resulting expression with the given options, we find that option D matches our result: A. V = 3 1 ( x 3 + 8 x 2 + 7 x ) B. V = x 2 + 8 x + 7 C. V = x 3 + 8 x 2 + 7 x D. V = 2 1 ( x 3 + 8 x 2 + 7 x )
Therefore, the correct answer is D.
Examples
Understanding the volume of prisms is crucial in various real-world applications. For instance, when designing a triangular support structure for a bridge, engineers need to calculate the volume of concrete required. Knowing the base and height of the triangular cross-section and the length of the support, they can use the formula V = 2 1 bh × l to determine the amount of material needed, ensuring structural integrity and cost-effectiveness.
