a rectangular patio is 13 feet by 84 feet. In walking from one corner of the patio to the opposite corner, how much distance is saved by cutting diagonally across the patio rather than walking around the outer edge?
Ahh -- time for the Pythagorean Theorem!
a^2 + b^2 = c^2
13^2 + 84^2 = c^2
169 + 7056 = c^2
7225 = c^2
85 = c
(13 + 84) - 85 = ___________ distance saved
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To determine how much distance is saved by cutting diagonally across the patio instead of walking around the outer edge, we need to compare the diagonal distance with the perimeter of the patio.
Step 1: Find the length of the diagonal using the Pythagorean theorem.
- The width of the patio is 13 feet, and the length is 84 feet.
- The diagonal can be found using the equation: diagonal^2 = width^2 + length^2.
- Substituting the given values, we get: diagonal^2 = 13^2 + 84^2.
- Simplifying, we have: diagonal^2 = 169 + 7056.
- Adding 169 and 7056 gives us: diagonal^2 = 7225.
- Taking the square root of both sides, we find: diagonal = √7225.
- Calculating the square root, we get: diagonal = 85 feet (rounded to the nearest foot).
Step 2: Calculate the perimeter of the patio and compare it to the diagonal length.
- The perimeter of a rectangle is given by the formula: perimeter = 2(width + length).
- Substituting the given values, we have: perimeter = 2(13 + 84) = 2(97) = 194 feet.
Step 3: Determine the distance saved by cutting diagonally.
- By walking around the outer edge, the distance covered would be equal to the perimeter, which is 194 feet.
- By cutting diagonally, the distance covered would be equal to the length of the diagonal, which is 85 feet.
- Therefore, the amount of distance saved by cutting diagonally is: Perimeter - Diagonal = 194 - 85 = 109 feet.
So, by cutting diagonally across the patio, you would save approximately 109 feet of walking distance compared to walking around the outer edge.
To find out how much distance is saved by cutting diagonally across the patio, we need to compare the distance of walking around the outer edge of the patio with the distance of walking diagonally across it.
First, let's calculate the distance of walking around the outer edge of the rectangular patio:
Since opposite sides of a rectangular shape are equal, we have two sides measuring 13 feet each and two sides measuring 84 feet each.
The distance of walking around the outer edge can be calculated by adding up the lengths of all four sides:
13 + 13 + 84 + 84 = 194 feet
Now, let's calculate the distance of walking diagonally across the patio:
This can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the two sides are 13 feet and 84 feet, and the hypotenuse is the distance we need to find.
Using the Pythagorean theorem, we can calculate the hypotenuse as follows:
hypotenuse = √(13^2 + 84^2)
hypotenuse = √(169 + 7056)
hypotenuse = √7225
hypotenuse ≈ 85 feet
Therefore, the distance of walking diagonally across the patio is approximately 85 feet.
To find the amount of distance saved by cutting diagonally, we subtract the distance of walking diagonally from the distance of walking around the outer edge:
Distance saved = Distance of walking around the outer edge - Distance of walking diagonally
Distance saved = 194 feet - 85 feet
Distance saved = 109 feet
So, cutting diagonally across the patio saves approximately 109 feet of distance.