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CCSS.Math:

the Nile River has overflowed and covered its entire surroundings except for the tip of the Great Pyramid in Giza Egypt an expedition was sent to find how high the water had risen the people measured the edge of the pyramid that's above the water and found it was 72 meters long so this distance right over here 72 meters they knew that the entire length of the edge is 180 meters when it's not flooded so this entire length is 180 meters they also knew that the vertical height of the pyramid is 139 meters so this is 139 meters what is the level of the water above the ground so the ground is right over here at the base of the pyramid and so they want the level of the water above the ground so that's this height this height right over here so let's just just call that H we want to figure out what H is round your answer if necessary to two decimal places so what do we know when what do we not know so they've labeled this little angle here theta and this of course is a right angle so this angle here at the base of the pyramid this is going to be the complement of theta it's going to be 90 degrees minus theta and using that information we can also figure out that this angle up here is also is also going to be theta if you don't if you don't if that looks a little bit strange to you let me just draw it out here and make it a little bit clearer make it a little bit clearer if we have a triangle right triangle where this angle right over here is 90 minus theta and we wanted to figure out what this is up here let's say this is X well we could say X plus 90 minus theta 90 degrees minus theta plus 90 degrees plus 90 degrees is going to be equal to well the sum of the of the angles of a triangle are going to be 180 degrees well if we subtract 180 from both sides so that's 100 and that's 100 that's 180 from the left hundred 80 from the right we get X minus theta is equal to zero or if you add theta to both sides X is equal to theta so this thing up here is going to be theta as well so this is also going to be theta and what else do we know well we know that this is 72 we know that the whole thing is 180 so this is 72 and the whole thing is 180 the part of this Edge's it's below the water this distance right over here let me draw it without cluttering the picture too much during the black color this distance right over here is going to be 108 108 plus 72 is 180 so what does this do it do for us we need to figure out this height we need to figure out this height we know that this right over here is a right triangle this right over here is a right triangle I could color this in just to make a little bit clearer this thing in yellow right over here is a right triangle if we look at that right triangle and if we wanted to solve for H and solve for H using a trig ratio based on this angle theta right over here we know that let's see relative to this angle theta relative this angle theta this side of length H is an adjacent side and this length of 108 right over here along the edge that's the hypotenuse of this yellow triangle that I just highlighted in so which trig ratio involves an adjacent side and a hypotenuse well we just write sohcahtoa so cut Toa sine is opposite over hypotenuse that would be this distance over the hypotenuse cosine is adjacent over hypotenuse adjacent over hypotenuse so we get the cosine of theta is going to be equal to the height that we care about that's the adjacent side of this right triangle over the length of the hypotenuse over 108 well that doesn't help us yet because we don't know what the cosine of theta is but there's a clue here theta is also sitting up here so maybe if we can figure out what cosine of theta is based up here then we can solve for H so if we look at this theta what is the cosine of theta and now we're looking at a different right triangle we're looking at this entire we're looking at this entire right triangle now based on that entire right triangle what is cosine of theta well cosine of theta once again is equal to adjacent over hypotenuse the adjacent length is this length right over here we already know that's 139 meters so it's going to be equal to 139 meters and what's the length of the hypotenuse well the hypotenuse is this leg right over here it's 72 72 plus 108 or we already have it labeled here it's 180 we can assume that this is an isosceles this this pyramid is an isosceles triangle so 180 on that side hundred 80 on that side so the cosine is adjacent 139 over the hypotenuse which is 180 over 180 and these are these data are the same data we just showed that so now we have cosine of theta is H over 108 cosine of theta is 139 over 180 or we could say that H over 108 which is cosine of theta also is equal to 139 over 180 both of these things are equal to cosine of theta now to solve for H we just multiply both sides by 108 so H is equal to 139 times 108 over 180 so let's get our calculator out and calculate that so that is going to be 139 times 108 divided by 180 gets us to 83 point four meters so H is equal to eighty three point four meters the height of the water is eighty three point four