To find how far from the house you should place the base of a 10-foot ladder when it's positioned at an angle of 75° from the ground, we can use trigonometry. Specifically, the cosine function is useful here, because it relates the adjacent side (distance from the house) to the hypotenuse (the length of the ladder).

The formula for cosine is:

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Here:

• $\theta = 75^\circ$
• Hypotenuse = 10 feet (length of the ladder)
• Adjacent = distance from the house (what we need to find)

Rearranging the formula to solve for the adjacent side:

$\text{adjacent} = \cos(75^\circ) \times 10$

Now, let's calculate:

$\text{adjacent} = \cos(75^\circ) \times 10 \approx 0.2588 \times 10 = 2.588 \text{ feet}$

So, the base of the ladder should be approximately 2.59 feet from the house.

Would you like more details or further explanation?

Here are 5 related questions:

1. How do you calculate the height the ladder reaches against the house?
2. How does the angle of the ladder affect its stability?
3. What if the ladder was 15 feet long but kept at the same angle?
4. Can you determine the required ladder length if you know the height and the angle?
5. What is the relation between sine, cosine, and the angles in a right triangle?

Tip: Using a larger angle (closer to 90°) means the ladder will be closer to the house, while a smaller angle makes it farther.