Question

a helium - filled balloon is attached to a long string and is at a height of 78 m from the ground. the string is tied to a hook on the ground and makes an angle of 30° with the ground, as shown below. diagram of a right - triangle with a 30° angle at the ground - hook, the height from the balloon to the ground is 78 m, and the string is the hypotenuse with a ? for its length what is the length of the string, in meters? options: 90.07 m, 78 m, 110.29 m, 156 m

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle where the height (opposite side to the 30° angle) is 78 m, and the string is the hypotenuse (\(c\)) we need to find. Using the sine function: \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)
\(\sin(30^\circ)=\frac{78}{c}\)

Step2: Solve for \(c\)

We know that \(\sin(30^\circ)=\frac{1}{2}\), so substitute:
\(\frac{1}{2}=\frac{78}{c}\)
Cross - multiply: \(c = 78\times2\)
\(c = 156\)

Answer:

156 m