Question

To calculate the volume of the described solid (pyramid).

Short answer

The volume of the pyramid is \(\frac{2}{3}{b^2}h\).

Step-by-step solution

Given data

The pyramid with height \(h\) and rectangular base with dimensions \(b\) and \(2b\).

Concept used of pyramid

A pyramid is a 3D figure built with a base of a polygon and triangular faces all connected together. A pyramid connects each vertex of the base to a common tip or apex gives it the typical shape. Let us learn more about pyramids in this section.

Solve to find the volume

Show the dimensions of the pyramid as in Figure 1.

Refer to Figure 1.

For the cross section, by the use of similar triangles for the base dimension of \(b\),

\(\begin{aligned}\frac{y}{b} &= \frac{x}{h}\\y &= \frac{b}{h}x\end{aligned}\)

The region lies between \(a = 0\) and \(b = h\).

The expression to find the volume of the pyramid as shown below.

\(V = \int_a^b A (x)dx\) …..(1)

Find the area of the pyramid as shown below.

\(\begin{aligned}A(x) &= 2y \times y\\ &= 2 \times \frac{b}{h}x \times \frac{b}{h}x\\ &= 2\frac{{{b^2}}}{{{h^2}}}{x^2}\end{aligned}\)

Substitute 0 for a, h for \(b\), and \(2\frac{{{b^2}}}{{{h^2}}}{x^2}\) for \(A(x)\) in Equation (1).

\(\begin{aligned}V &= \int_0^h 2 \frac{{{b^2}}}{{{h^2}}}{x^2}dx\\ &= 2\frac{{{b^2}}}{{{h^2}}}\left( {\frac{{{x^3}}}{3}} \right)_0^h\\ &= 2\frac{{{b^2}}}{{{h^2}}}\left( {\frac{{{h^3}}}{3} - 0} \right)\\ &= \frac{2}{3}{b^2}h\end{aligned}\)

Therefore, the volume of the pyramid is \(\frac{2}{3}{b^2}h\).