Find the height a 17 foot ladder can reach on the side of a building when it hits the ground at a `65^(@)` angle.

Options

A. 7.2 ft.
B. 15.4 ft.
C. 36.5 ft.
D. 48 ft.

Answer

To find the height that a 17-foot ladder can reach on the side of a building when it hits the ground at a 65-degree angle, we can use trigonometric functions. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The ladder forms a right triangle with the ground and the wall of the building. The length of the ladder is the hypotenuse of the triangle, and we need to find the height (the opposite side) when the ladder is positioned at a 65-degree angle with the ground.

### Step 2: Identify the Relevant Trigonometric Function
In a right triangle, the sine function relates the opposite side (height) to the hypotenuse (length of the ladder). The formula is:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Where:
- \(\theta\) is the angle with the ground (65 degrees).
- The opposite side is the height we want to find.
- The hypotenuse is the length of the ladder (17 feet).

### Step 3: Set Up the Equation
Using the sine function, we can express the height (h) as:
\[ h = L \cdot \sin(65^\circ) \]
Where \(L\) is the length of the ladder (17 feet).

### Step 4: Substitute the Values
Now, substitute the known values into the equation:
\[ h = 17 \cdot \sin(65^\circ) \]

### Step 5: Calculate \(\sin(65^\circ)\)
Using a calculator, we find:
\[ \sin(65^\circ) \approx 0.9063 \]

### Step 6: Calculate the Height
Now, substitute \(\sin(65^\circ)\) into the equation:
\[ h = 17 \cdot 0.9063 \approx 15.4 \text{ feet} \]

### Final Answer
The height that the ladder can reach on the side of the building is approximately **15.4 feet**.

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