To find how far the base of the ladder is from the house, we can use trigonometric functions. In this case, we will use the cosine function, which relates the adjacent side (the distance from the house) to the hypotenuse (the length of the ladder).
The formula using cosine is:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Where:
- \(\theta\) is the angle (68 degrees),
- the adjacent side is the distance from the base of the ladder to the house,
- the hypotenuse is the length of the ladder (12 feet).
Rearranging this formula to find the adjacent side gives us:
\[ \text{adjacent} = \text{hypotenuse} \times \cos(\theta) \]
Plugging in the values:
\[ \text{adjacent} = 12 \times \cos(68^\circ) \]
Now, we calculate \(\cos(68^\circ)\):
Using a calculator:
\[ \cos(68^\circ) \approx 0.3746 \]
Now we can calculate the distance:
\[ \text{adjacent} = 12 \times 0.3746 \approx 4.4952 \]
Rounding to the nearest tenth:
\[ \text{adjacent} \approx 4.5 \]
Thus, the base of the ladder is approximately 4.5 feet from the house.