Rapunzel is locked away in a 75 foot tower and has fashioned a crossbow and zip
ID: 3026257 • Letter: R
Question
Rapunzel is locked away in a 75 foot tower and has fashioned a crossbow and zip line (out of her hair) for her escape. She knows there is a 30-foot tree away from the base of the tower. If she wanted to create a zip line with the top of the tree, at what angle of depression (denoted theta) would she need to shoot headarrow and zip line and how long does the zip line need to be? Angle: ________ Length of Zipline: ____________ You stand at the bottom of a vertical cliff. At the top of the cliff grows a large tree and you discover that your line of sight to the top of the tree is 25.8 degree and your line of sight to the base of the tree is 19.6 degree. Your friend, who just climbed to the top of the cliff, estimates the cliff is 90 feet high. Use these facts to estimate the height of the tree (denoted by h in the diagram below) and your distance from the cliff. Height of tree: __________ Distance From Cliff: ____________Explanation / Answer
1) the height of tower - tree = ( 75 - 30 ) = 45 feet
applying tan rule
tan theta = 45 / 28
theta = tan^-1 ( 45/ 28 )
theta = 58.109 degrees
angle of depression is 58.11 degrees ( rounded to 2 decimal places )
length of zip
sin 58.109 = 45 / length of zip
length of zip = 45 / sin 58.109 = 53 feet
2) calculating base distance first
tan 19.6 = 84 / base
base = 84 / tan 19.6 = 235.90 feet
tan 25.8 = ( h+ 84 ) / 235.90
h + 84 = 114.038
h = 30.038
height of the tree is 30.04 feet approx
finding r
sin 19.6 = 84 / r
r = 84 / sin 19.6
r = 250.41 feet
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