To determine the angle of elevation formed between the board and the lower side of the ditch, we can use trigonometric principles. We know that the board forms the hypotenuse of a right triangle, with one leg being the height difference (3 feet) and the other leg being the horizontal distance across the ditch.
- Identify the components of the triangle:
- Hypotenuse (length of the board, \(c\)) = 6 feet
- Opposite side (height of the ditch, \(a\)) = 3 feet
- Adjacent side (horizontal distance, \(b\)) = to be determined
Using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ 6^2 = 3^2 + b^2 \]
\[ 36 = 9 + b^2 \]
\[ b^2 = 36 - 9 = 27 \]
\[ b = \sqrt{27} = 3\sqrt{3} \approx 5.196 \]
- Finding the angle of elevation: To find the angle of elevation (\(\theta\)), we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
\[ \sin(\theta) = \frac{a}{c} \]
Substituting the known values:
\[ \sin(\theta) = \frac{3}{6} = \frac{1}{2} \]
- Finding the angle using inverse sine: Now we can determine the angle using the inverse sine function:
\[ \theta = \arcsin\left(\frac{1}{2}\right) \]
From trigonometric values, we know that:
\[ \theta \approx 30^\circ \]
Thus, the approximate angle of elevation formed between the board and the lower side of the ditch is 30 degrees.
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