To find the distance the base of the ladder should be from the house, we need to use some basic trigonometry. When a ladder leans against a house, it forms a right triangle with the ground. The ladder acts as the hypotenuse, the distance from the base of the ladder to the house is the adjacent side, and the height where the ladder touches the house is the opposite side.

Given:

• Length of the ladder (hypotenuse) = 12 feet
• Angle between the ladder and the ground = 75°

We need to find the distance from the base of the ladder to the house (the adjacent side). We can use the cosine function, which relates the adjacent side and the hypotenuse in a right triangle:

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

Rearranging for the adjacent side:

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

Substituting the given values:

$\text{adjacent} = 12 \times \cos(75^\circ)$

Now, let's calculate it.The base of the 12-foot ladder should be approximately 3.1 feet away from the house to achieve the proper angle of 75° from the ground.

Would you like more details or have any questions?

Related Questions:

1. How high will the ladder reach on the wall at a 75° angle?
2. What happens to the distance from the base if the angle is reduced to 60°?
3. How would you calculate the height reached by the ladder using trigonometry?
4. What is the effect on stability if the ladder is placed at a steeper angle?
5. What are the safety considerations when setting up a ladder at different angles?

Tip: Always ensure that the ground is stable and the base of the ladder is on a non-slippery surface to avoid accidents.