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Problem 429

# A ladder, which is leaning against a building, makes an angle of $$75^{\circ}$$ with the ground. If the top of the ladder reaches a point which is 20 feet above the ground, find, to the nearest foot, the distance from the foot of the ladder to the base of the building.

### Short Answer

Expert verified
The distance from the foot of the ladder to the base of the building is approximately $$5$$ feet.
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## Step 1: Identify the given values and what we want to find

We are given: - Angle between ladder and ground: $$75^{\circ}$$ - Height from the ground to the top of the ladder: 20 feet We want to find the distance from the foot of the ladder to the base of the building.

## Step 2: Set up the equation using tangent function

In right triangle trigonometry, the tangent function is defined as the ratio of the opposite side to the adjacent side. Therefore, we can write the equation involving the tangent function as: $\tan(75^{\circ}) = \frac{\text{Opposite Side (height)}}{\text{Adjacent Side (distance)}}$ Where the Opposite Side (height) = 20 feet.

## Step 3: Solve for the Adjacent Side (distance)

Now, we want to solve for the Adjacent Side (distance), so let's rewrite the equation as: $\text{Adjacent Side (distance)} = \frac{\text{Opposite Side (height)}}{\tan(75^{\circ})}$ Now, plug in the given values and calculate the distance: $\text{Adjacent Side (distance)} = \frac{20}{\tan(75^{\circ})}$ $\text{Adjacent Side (distance)} \approx 5.213$

## Step 4: Round to the nearest foot

The distance is approximately 5.213 feet. Since we need to round it to the nearest foot, the final answer is: Distance from the foot of the ladder to the base of the building: $$5$$ feet.

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