To find the height of the entryway, we can use trigonometry. Specifically, we can use the sine function, which relates the angle of a right triangle to the ratio of the length of the opposite side (the height in this case) to the length of the hypotenuse (the walkway).
The formula for the sine of an angle is:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case:
- \(\theta = 56\) degrees
- \(\text{hypotenuse} = 11\) meters
- \(\text{opposite} = \text{height of the entryway}\)
Using the formula, we can rearrange it to solve for the height:
\[ \text{height} = \sin(56^\circ) \times \text{hypotenuse} \]
Now, let's calculate it:
\[ \text{height} = \sin(56^\circ) \times 11 \]
Using a calculator to find \(\sin(56^\circ)\):
\(\sin(56^\circ) \approx 0.8290\)
Now, substitute this value into the equation:
\[ \text{height} \approx 0.8290 \times 11 \approx 9.119 \]
Thus, the entryway is approximately 9.12 meters high.