Question

Oscar uses a board to form a walking plank across a small ditch that has uneven sides. The board is 8 ft in length and stretches across the entire gap. The higher side of the gap is 3 ft higher than the other side. What is the approximate angle of elevation formed between the board and the lower side of the ditch? Round your answer to the nearest whole number. (no decimals)

Answer 1

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:
\[
\boxed{22}
\]