2. A ladder of length 8 m rests against a wall so that the angle between the ladder and the wall is $$ 31^{\circ} $$. How far is the base of the ladder from the wall?

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The distance from the base of the ladder to the wall is approximately 6.86 meters. To find the distance from the base of the ladder to the wall, we can use trigonometry. Specifically, we can use the cosine function, which relates the angle, the adjacent side (the distance from the wall), and the hypotenuse (the length of the ladder). Given: - Length of the ladder (hypotenuse) = 8 m - Angle between the ladder and the wall = $$ 31^{\circ} $$ We can use the cosine function: $$ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} $$ In this case, the adjacent side is the distance from the base of the ladder to the wall, and the hypotenuse is the length of the ladder. Let $$ d $$ be the distance from the base of the ladder to the wall. Then we have: $$ \cos(31^{\circ}) = \frac{d}{8} $$ Now, we can solve for $$ d $$: $$ d = 8 \cdot \cos(31^{\circ}) $$ Using a calculator to find $$ \cos(31^{\circ}) $$: $$ \cos(31^{\circ}) \approx 0.8572 $$ Now, substituting this value back into the equation: $$ d \approx 8 \cdot 0.8572 \approx 6.8576 $$ Thus, the distance from the base of the ladder to the wall is approximately: $$ \boxed{6.86 \text{ m}} $$

2. A ladder of length 8 m rests against a wall so that the angle between the ladder and the wall is \( 31^{\circ} \). How far is the base of the ladder from the wall?

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2. A ladder of length 8 m rests against a wall so that the angle between the ladder and the wall is \( 31^{\circ} \). How far is the base of the ladder from the wall?

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