Question

Consider the image that shows a tower anchored by a guy with the guy wire is attached at point b on the tower 119 feet above the ground and then anchored to the ground at point a 120 feet from the base of the tower.

Options are: A) sin0= 119/120, B) sin0= 120/119, C) sin0= 119/169, and D) sin0= 120/169

Answer 1

Explanation:

`sin\theta=(opposite)/(hypotenuse)\\ \\ sin\theta=(119)/(169)\ (answer\ C)`

Answer 2

The ratio
`\sin \theta=(119)/(169)` is accurate for the described tower and guy-wire configuration.

Let's denote the angle between the tower and the guy wire as θ. The length of the guy wire is the hypotenuse of a right triangle formed by the tower, the ground, and the guy wire.

According to trigonometry, the sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height of the tower at point B (119 feet), and the hypotenuse is the length of the guy wire.

Therefore, the sine of angle θ (sin θ) is given by:

`\sin \theta=\frac{\text { Opposite }}{\text { Hypotenuse }}=\frac{119}{\text { length of guy wire }}`

Now, we need to find the length of the guy wire. This can be determined using the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):

`c^2=a^2+b^2`

In this case, a is the height of the tower at point B (119 feet), b is the distance from point A to the base of the tower (120 feet), and c is the length of the guy wire.

`\begin{aligned}& c^2=119^2+120^2 \\& c^2=14161+14400 \\& c^2=28561 \\& c=√(28561) \\& c=169\end{aligned}`

Now that we have the length of the guy wire (169 feet), we can substitute it back into the original expression for sin θ:

`\sin \theta=(119)/(169)`

Therefore, the correct answer is: C)
`\sin \theta=(119)/(169)`