To find the distance from the bottom of the ladder to the base of the wall, we can use trigonometry, specifically the cosine function. Here's how to solve the problem:

Step 1: Identify the Relevant Triangle

The ladder, the wall, and the ground form a right triangle. The ladder is the hypotenuse, the distance from the bottom of the ladder to the base of the wall is the adjacent side, and the height of the ladder against the wall is the opposite side.

Step 2: Apply the Cosine Function

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ Here:

• $\theta = 60^\circ$
• Hypotenuse = 18 feet

We need to find the length of the adjacent side, which is the distance from the bottom of the ladder to the base of the wall.

Step 3: Plug in the Values

$\cos(60^\circ) = \frac{\text{adjacent}}{18 \text{ feet}}$

We know that $\cos(60^\circ) = \frac{1}{2}$, so: $\frac{1}{2} = \frac{\text{adjacent}}{18}$

Step 4: Solve for the Adjacent Side

Multiply both sides by 18: $\text{adjacent} = 18 \times \frac{1}{2} = 9 \text{ feet}$

So, the distance from the bottom of the ladder to the base of the wall is 9 feet.

Do you want more details or have any questions?

Here are 8 related questions you might find helpful:

1. What is the sine of the angle and how can it help find the height of the ladder against the wall?
2. How would the distance change if the angle with the ground were smaller, say 45°?
3. How can we use the Pythagorean theorem to verify the distance found?
4. What if the ladder length was increased? How would it affect the distance?
5. Can we determine the height of the ladder against the wall using this information?
6. How would this problem change if the ladder formed an obtuse angle with the ground?
7. What other trigonometric functions are useful for solving right triangle problems?
8. How would you calculate the height if only the distance from the bottom to the wall was known?

Tip: For right triangles, trigonometric ratios like sine, cosine, and tangent are powerful tools for relating angles to side lengths.