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Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 6060 miles, and the eastbound airplane has traveled 8080 miles. How far apart are the two airplanes?

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Full solution

Q. Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 6060 miles, and the eastbound airplane has traveled 8080 miles. How far apart are the two airplanes?
  1. Identify Relationship: Identify the relationship between the distances traveled by the airplanes and the distance between them.\newlineThe paths of the airplanes form a right-angled triangle, with the distances traveled as the legs and the distance between them as the hypotenuse.
  2. Determine Leg Lengths: Determine the lengths of the legs of the triangle.\newlineNorthbound airplane: 6060 miles\newlineEastbound airplane: 8080 miles\newlineThese are the legs of the right-angled triangle.
  3. Apply Pythagorean Theorem: Apply the Pythagorean Theorem to find the distance between the airplanes.\newlineThe Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse cc is equal to the sum of the squares of the lengths of the other two sides aa and bb.\newlinea2+b2=c2a^2 + b^2 = c^2
  4. Solve for Distance: Plug in the known values into the Pythagorean Theorem and solve for cc. \newline(60 miles)2+(80 miles)2=c2(60 \text{ miles})^2 + (80 \text{ miles})^2 = c^2\newline3600+6400=c23600 + 6400 = c^2\newline10000=c210000 = c^2
  5. Find Square Root: Take the square root of both sides to find the value of cc.10000=c2\sqrt{10000} = \sqrt{c^2}100=c100 = c
  6. Conclude Final Answer: Conclude with the final answer.\newlineThe distance between the two airplanes is 100100 miles.

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