To solve this problem, we will use trigonometric ratios, specifically the sine function. Here’s the detailed solution:

Given:

1. The ladder forms an angle of $60^\circ$ with the ground.
2. The length of the ladder (hypotenuse) = $15 \, \text{feet}$.

We are tasked with finding how high up the wall the ladder reaches. Let the height up the wall be $h$.

Step 1: Use the sine ratio

The sine function relates the angle, opposite side, and hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Here:

• $\theta = 60^\circ$,
• Opposite = $h$ (height up the wall),
• Hypotenuse = $15 \, \text{feet}$.

Substitute into the equation: $\sin(60^\circ) = \frac{h}{15}$

Step 2: Value of $\sin(60^\circ)$

From trigonometric tables: $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$

Step 3: Solve for $h$

Substitute $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ into the equation: $\frac{\sqrt{3}}{2} = \frac{h}{15}$

Multiply both sides by $15$: $h = 15 \times \frac{\sqrt{3}}{2}$

Simplify: $h = \frac{15\sqrt{3}}{2}$

Using $\sqrt{3} \approx 1.732$: $h = \frac{15 \times 1.732}{2} \approx \frac{25.98}{2} \approx 13 \, \text{feet}.$

Final Answer:

The ladder reaches approximately 13 feet up the wall.

Do you want further details or have any questions?

Here are 5 related questions to practice:

1. A 20-foot ladder leans against a wall forming a $45^\circ$ angle with the ground. How high up does it reach?
2. A 10-foot ladder forms a $30^\circ$ angle with the ground. Find the height it reaches up the wall.
3. If a ladder reaches 12 feet up a wall and forms a $60^\circ$ angle with the ground, find the ladder's length.
4. A ladder makes a $75^\circ$ angle with the ground and reaches 18 feet up the wall. What is the ladder's length?
5. What is the angle formed by a 25-foot ladder if it reaches 20 feet up the wall?

Tip:

In trigonometric problems involving ladders, the height up the wall is always the opposite side relative to the angle with the ground. Use $\sin(\theta)$ to solve efficiently!