On a horizontal ground, the base of a straight ladder is 6 m away from the base of a vertical pole. The ladder makes an angle of 45° to the horizontal. If the ladder is resting at a point located at one-fifth of the height of the pole from the bottom, the height of the pole is ________ meters.
30
Understanding the Ladder and Pole Problem
This problem presents a scenario involving a straight ladder, a vertical pole, and the horizontal ground. It is a classic application of trigonometry, specifically dealing with right-angled triangles and trigonometric ratios to find unknown lengths when an angle and one side are known.
Visualizing the Setup
To solve this problem, it's helpful to visualize the situation as a right-angled triangle:
- The horizontal ground from the base of the pole to the base of the ladder forms one leg of the triangle.
- The vertical part of the pole, up to where the ladder touches it, forms the other leg.
- The ladder itself acts as the hypotenuse of this right-angled triangle.
The problem asks for the total height of the pole, given the distance from the ladder's base to the pole's base, the angle the ladder makes with the ground, and a crucial piece of information about where the ladder rests on the pole.
Key Information Provided
- The distance of the base of the straight ladder from the base of the vertical pole is \(\text{6 m}\). This is the adjacent side to the angle of elevation.
- The ladder makes an angle of \(\text{45}^\circ\) to the horizontal. This is the angle of elevation.
- The ladder is resting at a point located at one-fifth (\(\frac{\text{1}}{\text{5}}\)) of the total height of the pole from the bottom.
Step-by-Step Solution for Pole Height
1. Determining the Height the Ladder Reaches on the Pole
Let's denote the distance from the base of the ladder to the base of the pole as \(\text{d}\). We are given \(\text{d = 6 m}\).
Let \(\text{h}_\text{ladder}\) be the height on the pole where the ladder is resting. This is the opposite side to the angle of elevation in our right-angled triangle.
The angle the ladder makes with the horizontal ground is \(\theta = \text{45}^\circ\).
To relate the opposite side (\(\text{h}_\text{ladder}\)) and the adjacent side (\(\text{d}\)) with the angle (\(\theta\)), we use the tangent trigonometric ratio:
\(\text{tan}(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)
Substituting the given values into the formula:
\(\text{tan}(\text{45}^\circ) = \frac{\text{h}_\text{ladder}}{\text{6}}\)
We know that the value of \(\text{tan}(\text{45}^\circ)\) is \(\text{1}\).
\(\text{1} = \frac{\text{h}_\text{ladder}}{\text{6}}\)
To find \(\text{h}_\text{ladder}\), we multiply both sides of the equation by \(\text{6}\):
\(\text{h}_\text{ladder} = \text{1} \times \text{6}\)
\(\text{h}_\text{ladder} = \text{6 m}\)
So, the ladder touches the pole at a height of 6 meters from the horizontal ground.
2. Calculating the Total Height of the Pole
The problem provides a crucial relationship: the point where the ladder rests is at one-fifth (\(\frac{\text{1}}{\text{5}}\)) of the total height of the pole from the bottom.
Let \(\text{H}_\text{pole}\) represent the total height of the pole.
According to the problem statement, we can write the relationship as:
\(\text{h}_\text{ladder} = \frac{\text{1}}{\text{5}} \times \text{H}_\text{pole}\)
We have already calculated \(\text{h}_\text{ladder} = \text{6 m}\). Now we substitute this value into the equation:
\(\text{6} = \frac{\text{1}}{\text{5}} \times \text{H}_\text{pole}\)
To solve for \(\text{H}_\text{pole}\), we multiply both sides of the equation by \(\text{5}\):
\(\text{H}_\text{pole} = \text{6} \times \text{5}\)
\(\text{H}_\text{pole} = \text{30 m}\)
Thus, the total height of the pole is 30 meters.
Final Answer Summary
The problem was solved by first utilizing the given horizontal distance and the angle of elevation of the ladder to determine the height at which the ladder made contact with the pole. Then, using the information that this contact height represented one-fifth of the pole's total height, the full height of the pole was calculated.
| Parameter | Value |
|---|---|
| Distance from pole base to ladder base | \(\text{6 m}\) |
| Angle of ladder with horizontal ground | \(\text{45}^\circ\) |
| Height ladder reaches on pole (\(\text{h}_\text{ladder}\)) | \(\text{6 m}\) |
| Fraction of pole height at which ladder rests | \(\frac{\text{1}}{\text{5}}\) |
| Total height of the pole (\(\text{H}_\text{pole}\)) | \(\text{30 m}\) |
The height of the pole is 30 meters.
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