Q. 59 - Calculus

Question

Rebecca plans to visit the Great Pyramid of Giza in Cairo, Egypt. The Great Pyramid has a square base with each side approximately 756 feet in length. 

(a) An interesting fact about the Great Pyramid of Giza is that the perimeter of its base is equal to the circumference of a circle whose radius is equal to the height of the pyramid. Use this fact to find the height of the Great Pyramid, rounded to the nearest integer.

 (b) The volume of a square-based pyramid with base area A and height h is given by the formula $V=\frac{1}{3}Ah$. Use this formula to find the volume of the Great Pyramid. 

(c) Find the volume of the Great Pyramid in another, much more difficult way: Set up a definite integral that represents the volume, and solve it. Use the given diagram as a starting point, and justify your answer. 

Step-by-Step Solution

Answer

a) The height of the pyramid is $481$ feet.

b)The Volume of the pyramid is $91636272$ cubic feet.

c)The volume of the pyramid is $91636272$ cubic feet.

Step 1. Given information

Great Pyramid of Giza has square base of side-length$756$ feet.

The perimeter of the base of pyramid is equal to the circumference of a circle whose radius is the height of the pyramid.

Part (a) Step 1. Find height of the pyramid.

The square base has perimeter :

$4\times 756=3024$ feet.

Since perimeter is the circumference of a circle of radius is the height of pyramid:

So according to the formula of circumference of circle,

$$\begin{gathered} 2\mathrm{\pi h}=3024 \\ \mathrm{h}=\frac{3024}{2\mathrm{\pi }} \\ \mathrm{h}=\frac{3024}{2\times {\displaystyle \frac{22}{7}}} \\ \mathrm{h}\approx 481 \end{gathered}$$

so height of the pyramid is approximately $481$ feet.

Part (b) Step 1. To find volume of the pyramid.

The volume of the pyramid is $V=\frac{1}{3}Ah$, where $A$ is the area of square base.

So $$\begin{gathered} A={\left(756\right)}^2 \\ =571536 \end{gathered}$$

Hence volume of the pyramid is:

$$\begin{gathered} V=\frac{1}{3}\times 571536\times 481 \\ =91636272 \end{gathered}$$

So the volume of the pyramid is $91636272$ cubic feet.

Part (c) Step 1. To find volume by using integration:

Cut a slice of the pyramid whose thickness is 

In the above figure we can see that $∆ABD~∆AOC$

So,

$$\begin{gathered} \frac{s}{756}=\frac{x}{481} \\ s=\frac{756}{481}x \end{gathered}$$

Area of slice:

$$\begin{gathered} A=s^2 \\ ={\left(\frac{756}{481}x\right)}^2 \\ ={\left(\frac{756}{481}\right)}^2{x}^2 \end{gathered}$$

Volume of slice:

$dv={\left(\frac{756}{481}\right)}^2{x}^{2}dx$

Hence volume of the full pyramid is:

$$\begin{gathered} v={\int }_0^{481}dv \\ ={\int }_0^{481}{\left(\frac{756}{481}\right)}^2{x}^{2}dx \\ ={\left(\frac{756}{481}\right)}^2{\int }_0^{481}{x}^{2}dx \\ ={\left(\frac{756}{481}\right)}^2{{\left[\frac{x^3}{3}\right]}^{481}}_0 \\ ={\left(\frac{756}{481}\right)}^2\left[\frac{{481}^3}{3}-0\right] \\ =\frac{1}{3}\times {\left(756\right)}^2\times \left(481\right) \\ =91636272 \end{gathered}$$

So the volume of the pyramid is $91636272$cubic feet.