Question

a solid oblique pyramid has a square base with edges measuring x cm. the height of the pyramid is (x + 2) cm. which expression represents the volume of the pyramid? \\(\frac{x^3 + 2x^2}{3}\\) cm³ \\(\frac{x^2 + 2x^2}{2}\\) cm³ \\(\frac{x^3}{3}\\) cm³ \\(\frac{x^3 + 2x^2}{2}\\) cm³

Explanation:

Step1: Recall the volume formula for a pyramid

The volume \( V \) of a pyramid is given by \( V=\frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height.

Step2: Calculate the area of the square base

The base is a square with edge length \( x \) cm. The area of a square is \( B = \text{side}^2 \), so \( B=x^2 \) \( \text{cm}^2 \).

Step3: Substitute \( B \) and \( h \) into the volume formula

The height \( h=(x + 2) \) cm. Substituting \( B = x^2 \) and \( h=(x + 2) \) into \( V=\frac{1}{3}Bh \), we get:
\[
V=\frac{1}{3}\times x^2\times(x + 2)
\]

Step4: Expand the expression

Using the distributive property \( a(b + c)=ab+ac \), where \( a = x^2 \), \( b=x \), and \( c = 2 \), we have \( x^2\times(x + 2)=x^3+2x^2 \). So the volume is:
\[
V=\frac{x^3 + 2x^2}{3}\text{ cm}^3
\]

Answer:

\(\frac{x^{3}+2x^{2}}{3}\text{ cm}^3\) (the first option)