Question

a tree needs to be staked down before a storm. if the ropes can be tied on the tree trunk 8 feet above the ground and the staked rope should make a 30 degree angle with the ground, how far, to the nearest tenth, from the base of the tree should each rope be staked? 11.3 13.9 4.6 5.7

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle situation. The height on the tree trunk (opposite side) is \(h = 8\) feet, and the angle with the ground is \(\theta=30^{\circ}\). We want to find the distance from the base of the tree (adjacent side), let's call it \(x\). We use the tangent function: \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Substitute the values into the tangent formula

Since \(\tan\theta=\frac{h}{x}\), and \(\theta = 30^{\circ}\), \(h = 8\). We know that \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\approx0.577\). So, \(x=\frac{h}{\tan\theta}\).

Step3: Calculate the value of \(x\)

Substitute \(h = 8\) and \(\tan30^{\circ}\approx0.577\) into \(x=\frac{h}{\tan\theta}\). Then \(x=\frac{8}{\tan30^{\circ}}=\frac{8}{\frac{1}{\sqrt{3}}}=8\sqrt{3}\approx8\times1.732 = 13.856\approx13.9\)

Answer:

13.9