To find the angle between the side of the house (the vertical side) and the cat's line of sight, we can use the sine function. We know that the opposite side (the height of the cat above the ground) is the height of the house, which we will denote as \( h \), and the hypotenuse (the diagonal distance from the cat to you) is 18 feet.
We also know the distance from you to the base of the house (the adjacent side) is 12 feet. The angle \( θ \) we want to find is the angle formed by the line of sight of the cat and the vertical side of the house.
Using the formula for sine: \[ \sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{18} \]
We also have to find \( h \) using the Pythagorean theorem, where the full triangle consists of the height of the house and the distance you are from the house: \[ h^2 + 12^2 = 18^2 \]
Calculating \( 12^2 \) and \( 18^2 \): \[ h^2 + 144 = 324 \] \[ h^2 = 324 - 144 \] \[ h^2 = 180 \] \[ h = \sqrt{180} ≈ 13.42 \]
Now, substituting this value into the sine ratio: \[ \sin(θ) = \frac{13.42}{18} \] \[ \sin(θ) ≈ 0.746 \]
Using the inverse sine function to find \( θ \): \[ θ = \arcsin(0.746) \]
Calculating \( θ \): \[ θ ≈ 48.5 \text{ degrees} \]
Since the angle we need to find is between the vertical and the line of sight of the cat, we can also find the corresponding angle using: \[ 40 \text{ degrees} = 90 - 48.5 \]
Rounding to the nearest whole degree gives us: \[ θ ≈ 41 \text{ degrees} \]
Thus, the correct response is: 41 degrees
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