We can solve this problem by using trigonometry, particularly the tangent function. Here's how:

Problem Breakdown:

• The tree is 50 inches tall.
• The rope is tied exactly halfway up the tree, meaning the rope's height from the ground is half of 50 inches, which is 25 inches.
• The rope forms a 55° angle with the ground.
• We need to find the horizontal distance (base of the tree to the rope stake in the ground).

Step-by-Step Solution:

Let’s label the knowns:

• Opposite side (height of the rope): $25 \, \text{inches}$
• Angle: $55^\circ$
• Adjacent side (horizontal distance): This is what we are trying to find.

Using the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Substituting the known values: $\tan(55^\circ) = \frac{25}{\text{adjacent}}$ Now solve for the adjacent side: $\text{adjacent} = \frac{25}{\tan(55^\circ)}$

Let me calculate that for you.The distance from the base of the tree to the point where the rope is staked into the ground is 17.5 inches.

Would you like further details or have any other questions?

Here are some related questions to expand the concept:

1. How would the solution change if the rope was tied at a different height?
2. What trigonometric function would you use if the angle and the adjacent side were given instead?
3. Can you calculate the length of the rope if it forms the hypotenuse of a right triangle?
4. How would the answer differ if the angle was smaller, say 45°?
5. Could you apply the same principles to find the distance for a taller tree?

Tip: When using trigonometric functions, always make sure your calculator is in the correct mode (degrees or radians) depending on the angle you're working with.