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

16cm3

Explanation:

Answer 2

The volume of this pyramid is 16 cm³.

Explanation:

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

`\displaystyle V = (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 (x)/(3)* 4 = (4)/(3)x`.

The area of the square will be

`\displaystyle \left((4)/(3)x\right)^(2) = (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){(16)/(9)x^(2)\cdot dx} &= (16)/(9)\int_(0)^(3){x^(2)\cdot dx}\\ &= (16)/(9)\cdot \left((1)/(3)\int_(0)^(3){3x^(2)\cdot dx}\right) & \text{Set up the integrand for power rule}\\ &= \left.(16)/(9)* (1)/(3)\cdot x^(3)\right|^(3)_(0)\\ &= (16)/(27)* 3^(3) \\ &= 16\end{aligned}`.

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