One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about 2580 BC. Its original height was 480 feet 11 inches, but owing to the loss of it's topmost stones, it is now shorter. Find the current height of the Great Pyramid using the information given in the illustration.

The entire paragraph of text given to you is not needed. It's interesting that the height used to be 480 ft 11 in, but we don't use that anywhere in our calculations to get the current shorter height. Instead we rely solely on the diagram.

Refer to the diagrams I have posted below. Figure 1 is the same diagram more or less, but I've added the variables x and h.

x = horizontal distance from the center of the pyramid to the left most part of the pyramid.
h = current height of the pyramid

Both h and x are in feet.

I've also added the point labels A through E.

I've split figure 1 into two pieces. Figure 2 focuses on triangle ADE and figure 3 focuses on triangle BDE.

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Let's put our attention on triangle ADE in figure 2.

Set up the tangent ratio shown below and solve for h.

[tex]\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\\\\\tan(\text{A}) = \frac{\text{DE}}{\text{AD}}\\\\\tan(40.3) = \frac{h}{x+200}\\\\h = (x+200)\tan(40.3)\\\\[/tex]

Now move onto triangle BDE in figure 3.

Set up the tangent ratio and apply substitution.

[tex]\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}\\\\\tan(\text{B}) = \frac{\text{DE}}{\text{BD}}\\\\\tan(46.27) = \frac{h}{x+100}\\\\\tan(46.27) = \frac{(x+200)\tan(40.3)}{x+100}\\\\[/tex]

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Let's solve for x.

[tex]\tan(46.27) = \frac{(x+200)\tan(40.3)}{x+100}\\\\(x+100)\tan(46.27) = (x+200)\tan(40.3)\\\\x\tan(46.27)+100\tan(46.27) = x\tan(40.3)+200\tan(40.3)\\\\[/tex]

[tex]x\tan(46.27)-x\tan(40.3)=200\tan(40.3)-100\tan(46.27)\\\\x(\tan(46.27)-\tan(40.3))=200\tan(40.3)-100\tan(46.27)\\\\x=\frac{200\tan(40.3)-100\tan(46.27)}{\tan(46.27)-\tan(40.3)}\\\\x\approx 329.87238475414\\\\[/tex]

Use this to find h.

[tex]h = (x+200)\tan(40.3)\\\\h \approx (329.87238475414+200)\tan(40.3)\\\\h \approx 449.364461010933\\\\h\approx 449.364[/tex]

The height of the pyramid is currently 449.364 feet approximately.

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Lastly we'll convert to feet,inches format since the "480 ft 11 in" figure is in that format.

h = 449.364 ft

h = 449 ft + 0.364 ft

h = 449 ft + (0.364 ft)*(12 in/1 ft)

h = 449 ft + 4.368 in

h = 449 ft + 4 in

h = 449 feet, 4 inches