Question

A solid right pyramid has a square base. The length of the base edge is4 cm and the height of the pyramid is 3 cm period what is the volume of the pyramid?

Answer 1

Answer:

The volume of this pyramid is 16 cm³.

Step-by-step explanation:

The volume $V$ of a solid pyramid can be given as:

$\displaystyle V = \frac{1}{3} \cdot b \cdot h$,

where

• $b$ is the area of the base of the pyramid, and
• $h$ is the height of the pyramid.

Here's how to solve this problem with calculus without using the previous formula.

Imaging cutting the square-base pyramid in half, horizontally. Each horizontal cross-section will be a square. The lengths of these squares' sides range from 0 cm to 3 cm. This length will be also be proportional to the vertical distance from the vertice of the pyramid.

Refer to the sketch attached. Let the vertical distance from the vertice be $x$ cm.

• At the vertice of this pyramid, $x = 0$ and the length of a side of the square is also $0$.
• At the base of this pyramid, $x = 3$ and the length of a side of the square is $4$ cm.

As a result, the length of a side of the square will be

$\displaystyle \frac{x}{3}\times 4 = \frac{4}{3}x$.

The area of the square will be

$\displaystyle \left(\frac{4}{3}x\right)^{2} = \frac{16}{9}x^{2}$.

Integrate the area of the horizontal cross-section with respect to $x$

• from the top of the pyramid, where $x = 0$,
• to the base, where $x = 3$.

$\displaystyle \begin{aligned}\int_{0}^{3}{\frac{16}{9}x^{2}\cdot dx} &= \frac{16}{9}\int_{0}^{3}{x^{2}\cdot dx}\\ &= \frac{16}{9}\cdot \left(\frac{1}{3}\int_{0}^{3}{3x^{2}\cdot dx}\right) & \text{Set up the integrand for power rule}\\ &= \left.\frac{16}{9}\times \frac{1}{3}\cdot x^{3}\right|^{3}_{0}\\ &= \frac{16}{27}\times 3^{3} \\ &= 16\end{aligned}$.

In other words, the volume of this pyramid is 16 cubic centimeters.