Question

question 5 (multiple choice worth 4 points)
(05.01 mc)
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.
what is the length of the string, in meters?
○ 90.07 m
○ 78 m
○ 110.28 m
○ 136 m

Explanation:

Step1: Identify the trigonometric relationship

We know the height (opposite side to the \(30^{\circ}\) angle) is \(78\) m and we need to find the length of the string (hypotenuse \(y\)). Using the sine function: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), so \(\sin30^{\circ}=\frac{78}{y}\).

Step2: Solve for \(y\)

Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{78}{y}\). Cross - multiplying gives \(y = 78\times2\).

Answer:

\(156\) m