To determine how high up the tree the ladder will reach, we can use some trigonometric principles. Here's a detailed, step-by-step solution:
### Step 1: Understand the Problem
We are given:
- The distance from the base of the tree to the point where the ladder touches the ground (base of the right triangle) is 8 feet.
- The angle formed by the ladder with the ground is 57 degrees.
### Step 2: Identify Relevant Trigonometric Function
We need to find the vertical height (opposite side) of the right triangle formed by the ladder, which touches the tree.
The tangent function is particularly useful here because it relates the angle of a right triangle to the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Where:
- [tex]\(\theta\)[/tex] is the angle,
- The "opposite" side is the height we need to find,
- The "adjacent" side is the base (8 feet).
### Step 3: Set Up the Equation
Given:
- Adjacent side = 8 feet,
- [tex]\(\theta\)[/tex] = 57 degrees,
Our equation becomes:
[tex]\[
\tan(57^\circ) = \frac{\text{height}}{8}
\][/tex]
### Step 4: Solve for the Height
To get the height:
[tex]\[
\text{height} = 8 \times \tan(57^\circ)
\][/tex]
### Step 5: Calculate [tex]\(\tan(57^\circ)\)[/tex]
Using a calculator or trigonometric table to find [tex]\(\tan(57^\circ)\)[/tex]:
[tex]\[
\tan(57^\circ) \approx 1.53986
\][/tex]
### Step 6: Multiply to Find the Height
Now, multiply the tangent value by the adjacent side:
[tex]\[
\text{height} = 8 \times 1.53986 \approx 12.31892
\][/tex]
### Step 7: Round the Result
Finally, round the height to the nearest tenth:
[tex]\[
\text{height} \approx 12.3 \text{ feet}
\][/tex]
### Conclusion
The ladder will reach approximately 12.3 feet up the tree.
