To find the angle of elevation formed between the board and the lower side of the ditch, we can use trigonometry.
Given:
- Length of the board (hypotenuse, \(h\)): 8 ft
- Height difference (opposite side, \(a\)): 3 ft
- Base (adjacent side, \(b\)) can be found using the Pythagorean theorem.
Using the Pythagorean theorem: \[ h^2 = a^2 + b^2 \] Substituting the values we have: \[ 8^2 = 3^2 + b^2 \] \[ 64 = 9 + b^2 \] \[ b^2 = 64 - 9 = 55 \] \[ b = \sqrt{55} \approx 7.416 \]
Now, we need to find the angle of elevation (\(\theta\)) using the sine function: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{h} = \frac{3}{8} \]
To find \(\theta\), take the inverse sine: \[ \theta = \arcsin\left(\frac{3}{8}\right) \]
Calculating \(\theta\): \[ \theta \approx 22.03^\circ \]
Rounding this to the nearest whole number, the angle of elevation formed between the board and the lower side of the ditch is: \[ \boxed{22} \]
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