A 20ft long ladder is resting against a building. The ladder forms an angle of 50 degrees with the ground. How far from the building is the base of the ladder in feet?

We can solve this problem using trigonometry. Specifically, we will use the cosine function, which relates the adjacent side (distance from the base of the ladder to the building) and the hypotenuse (the length of the ladder) in a right triangle.

### Cosine formula:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Where:
- \(\theta = 50^\circ\)
- The hypotenuse (the ladder length) = 20 ft
- The adjacent side = the distance from the base of the ladder to the building (what we're solving for)

Now, rearrange the formula to solve for the adjacent side:
\[
\text{adjacent} = \text{hypotenuse} \times \cos(\theta)
\]
\[
\text{adjacent} = 20 \times \cos(50^\circ)
\]

Now, let's calculate this:

\[
\text{adjacent} = 20 \times 0.6428 \approx 12.86 \, \text{ft}
\]

### Therefore, the base of the ladder is approximately 12.86 feet from the building.

Would you like more details or have any other questions?

Here are some related questions to explore:
1. What is the height of the ladder against the building?
2. How would the distance from the building change if the ladder formed a 60-degree angle?
3. What is the length of the shadow cast by the ladder if the sun is directly overhead?
4. How would the problem change if the ladder is 30 feet long instead of 20 feet?
5. Can we calculate the area of the triangle formed by the ladder, the ground, and the building?

**Tip:** In right-angle triangles, the sine function gives the opposite side when the angle and hypotenuse are known, while cosine gives the adjacent side.

Learn how to calculate the distance from a building to the base of a 20ft ladder using the cosine function. The ladder forms a 50-degree angle with the ground, and we use trigonometric principles to find the distance. Ideal for students in Grades 9-11.

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