Answers:
- Height = [tex]3\sqrt{15}[/tex] feet
- Distance they run = 150 meters
- Exactly 9 feet away from the base.
- Exact length is [tex]5\sqrt{13}[/tex] feet
The diagrams for each are shown below (attached image).
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Work Shown:
Problem 1
[tex]a^2 + b^2 = c^2\\\\3^2 + b^2 = 12^2\\\\9 + b^2 = 144\\\\b^2 = 144 - 9\\\\b^2 = 135\\\\b = \sqrt{135}\\\\b = \sqrt{9*15}\\\\b = \sqrt{9}*\sqrt{15}\\\\b = 3\sqrt{15}\\\\[/tex]
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Problem 2
[tex]a^2 + b^2 = c^2\\\\90^2 + 120^2 = c^2\\\\8100 + 14400 = c^2\\\\22500 = c^2\\\\c^2 = 22500\\\\c = \sqrt{22500}\\\\c = 150\\\\[/tex]
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Problem 3
[tex]a^2 + b^2 = c^2\\\\a^2 + 12^2 = 15^2\\\\a^2 + 144 = 225\\\\a^2 = 225 - 144\\\\a^2 = 81\\\\a = \sqrt{81}\\\\a = 9\\\\[/tex]
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Problem 4
[tex]a^2 + b^2 = c^2\\\\10^2 + 15^2 = c^2\\\\100 + 225 = c^2\\\\325 = c^2\\\\c^2 = 325\\\\c = \sqrt{325}\\\\c = \sqrt{25*13}\\\\c = \sqrt{25}*\sqrt{13}\\\\c = 5\sqrt{13}\\\\[/tex]
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Side notes:
- When simplifying the square root, the idea is to pull out the largest perfect square factor if possible. Then I used the rule that [tex]\sqrt{x*y} = \sqrt{x}*\sqrt{y}[/tex] to break up the roots.
- I have 'a' as the horizontal leg and b as the vertical leg in each of the drawings below. The order doesn't matter but I did this to stay consistent.
- When applying the pythagorean theorem [tex]a^2+b^2 = c^2[/tex], the value of c is always the longest side (aka hypotenuse). This side is always opposite the 90 degree angle.
