Chapter 8.2, Problem 53AYU

To determine

Find the current height of the Great Pyramid using the information given in the illustration.

Answer to Problem 53AYU

  $\boxed{449.36ft}$

Explanation of Solution

Given information:

One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about $\text{258}{0}_{BC}$ . Its original height was $480$ feet $11$ inches, but owing to the loss of its top most stones, it is now shorter . Find the current height of the Great Pyramid using the information given in the illustration.

Source: Guinness Book of World Records.

  

Calculation:

Let the current height of the pyramid be $h$ .

  

From the figure ,it is clear that the angle C is $180°-46.27°=133.37°$ .

We have $A+40.3°+C=180°$ .

Substitute $133.37°$ for $C$ .

  $A+40.3°+133.37=180°$

  $A+174.03°=180°$

Subtract $174.03°$ from both the sides.

  $A+174.03°-174.03°=180°-164.03°$

  $A=5.97°$

From the figure, we can say that the length of the side opposite to angle $A$ is

  $200ft-100ft=100ft$

Use the Law of Sines to find the hypotenuse of the larger triangle.

  $\frac{\sin 5.97°}{100}=\frac{\sin 133.73°}{\text{hypotenuse}}$

  $0.00104=\frac{\sin 133.73°}{\text{Hypotenuse}}$

Multiply both the sides by $\frac{\text{Hypotenuse}}{0.00104}$

  $0.00104\left(\frac{\text{Hypotenuse}}{0.00104}\right)=\frac{\sin 133.73°}{hypotenuse}\left(\frac{\text{Hypotenuse}}{0.00104}\right)$

  $hypotenuse=694.812$

Consider the larger triangle with height $h$ .

  $\sin 40.3°=\frac{h}{694.812}$

Multiply both the sides by $694.812$ .

  $698.812\left(\sin 40.3°\right)=\frac{h}{694.812}\left(694.812\right)$

  $449.36=h$

Hence , the current height of the pyramid is $449.36ft$ .