To find a possible measurement for the horizontal distance of the triangle formed by the ramp, you can use the Pythagorean theorem. The ramp forms a right triangle with the vertical rise (height) and horizontal distance (base) as the two legs of the triangle, and the length of the ramp as the hypotenuse.
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Calculating the Horizontal Distance:
- Let \( h \) be the vertical rise (14 inches) and \( d \) be the horizontal distance.
- The length of the ramp (hypotenuse) is 16 inches. According to Pythagorean theorem: \[ d^2 + h^2 = \text{ramp length}^2 \] \[ d^2 + 14^2 = 16^2 \] \[ d^2 + 196 = 256 \] \[ d^2 = 256 - 196 \] \[ d^2 = 60 \] \[ d = \sqrt{60} \approx 7.75 \text{ inches} \] So, a possible measurement for the horizontal distance is approximately 7.75 inches.
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Finding the Third Angle of the Triangle:
- In a right triangle, one angle is 90°. Given that the ramp angle is 4.75°, you can find the third angle \( \theta \): \[ \text{Third angle} = 90° - \text{ramp angle} = 90° - 4.75° = 85.25° \]
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Identifying the Shortest Side of the Triangle:
- In a right triangle, the shortest side is usually the vertical rise (height) or the horizontal distance (base). In this scenario, since the height is 14 inches and the horizontal distance calculated is approximately 7.75 inches, the horizontal distance is shorter, making it the shortest side of the triangle.
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Evaluating possible horizontal distance for the second ramp:
- For a ramp with a length of 3 feet (which is 36 inches) and a vertical rise of 20 inches, we can check if a horizontal distance of 16 inches is possible using the Pythagorean theorem: \[ d^2 + h^2 = \text{ramp length}^2 \] \[ 16^2 + 20^2 = 36^2 \] \[ 256 + 400 = 1296 \] \[ 656 \neq 1296 \] Since 656 is not equal to 1296, it is not possible to have a horizontal distance of 16 inches given the specified ramp length and vertical rise. The correct horizontal distance needs to satisfy the Pythagorean theorem, which in this case would yield an amount less than 16 inches.
This analysis confirms whether specified dimensions are physically possible and meets required regulations for accessibility.