On a plane area there are two vertical towers separated by 100 feet apart. The shorter tower is 40 feet tall. A pole of length 6 feet stands on the line joining the base of two towers so that the tip of the towers and tip of the pole are also on the same line. If the distance of the pole from the shorter tower is 75 feet, then what is the height of the taller tower (approximately)?
85 feet
Detailed Solution: Finding the Taller Tower Height
This problem requires us to calculate the height of a taller vertical tower. We are given the height of a shorter tower, the distance between them, and information about a pole positioned such that its tip is collinear with the tips of both towers. This setup strongly suggests the use of similar triangles or coordinate geometry.
Understanding the Geometric Setup
Let's list the known values and what we need to find:
- Height of the shorter tower (\(h_1\)): 40 feet
- Height of the taller tower (\(h_2\)): Unknown
- Height of the pole (\(h_p\)): 6 feet
- Distance between the bases of the two towers: 100 feet
- Distance of the pole's base from the shorter tower's base: 75 feet
The key condition is that the tips of the two towers and the tip of the pole lie on the same straight line. This collinearity is the basis for our calculations.
Analyzing the Arrangement using Coordinate Geometry
To solve this, we can set up a coordinate system. Let the ground be the x-axis.
Assume the base of the shorter tower (B1) is at the origin (0, 0). Its tip (T1) is therefore at coordinates (0, 40).
The base of the taller tower (B2) is 100 feet away from B1. Let's place it at (100, 0). Its tip (T2) is at (100, \(h_2\)).
Now consider the pole. Its base (P) is 75 feet from B1. There are three possibilities for the pole's location relative to B1:
- Pole between towers: P is at (75, 0). Tip_p is at (75, 6). The collinear points are T1(0, 40), Tip_p(75, 6), and T2(100, \(h_2\)). The slope from T1 to Tip_p is \(\frac{6 - 40}{75 - 0} = \frac{-34}{75}\). Since the slope must be constant, the slope from Tip_p to T2 is \(\frac{h_2 - 6}{100 - 75} = \frac{h_2 - 6}{25}\). Equating slopes gives \(\frac{-34}{75} = \frac{h_2 - 6}{25}\), leading to \(h_2 = 6 - \frac{34}{3} \approx -5.33\). A negative height is impossible, and it contradicts \(h_2\) being the height of the *taller* tower (it should be > 40).
- Pole before the shorter tower: P is at (-75, 0). Tip_p is at (-75, 6). The collinear points are Tip_p(-75, 6), T1(0, 40), and T2(100, \(h_2\)). This configuration forms an upward sloping line.
- Pole after the taller tower: P is at (100 + 75, 0) = (175, 0). Tip_p is at (175, 6). The collinear points are T1(0, 40), T2(100, \(h_2\)), and Tip_p(175, 6). The slope from T1 to T2 is \(\frac{h_2 - 40}{100}\). The slope from T2 to Tip_p is \(\frac{6 - h_2}{175 - 100} = \frac{6 - h_2}{75}\). Equating slopes gives \(\frac{h_2 - 40}{100} = \frac{6 - h_2}{75}\), leading to \(7h_2 = 144\), so \(h_2 = \frac{144}{7} \approx 20.57\). This height is less than the shorter tower's height (40 feet), contradicting the description of \(h_2\) as the height of the *taller* tower.
Therefore, the only plausible arrangement is Case 2, where the pole is located 75 feet before the shorter tower.
Calculating the Taller Tower Height
Using the coordinates from Case 2:
- Tip of the pole (Tip_p): (-75, 6)
- Tip of the shorter tower (T1): (0, 40)
- Tip of the taller tower (T2): (100, \(h_2\))
Since these points are on the same line, the slope between Tip_p and T1 must be equal to the slope between T1 and T2.
Slope Calculation
The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: $\(m = \frac{y_2 - y_1}{x_2 - x_1}$\)
Calculate the slope between Tip_p(-75, 6) and T1(0, 40):
$\(m = \frac{40 - 6}{0 - (-75)} = \frac{34}{75}$\)Calculate the slope between T1(0, 40) and T2(100, \(h_2\)):
$\(m = \frac{h_2 - 40}{100 - 0} = \frac{h_2 - 40}{100}$\)Equating Slopes and Solving for \(h_2\)
Now, we equate the two slope expressions:
$\( \frac{34}{75} = \frac{h_2 - 40}{100} $\)Solve for \(h_2\):
Multiply both sides by 100:
$\( h_2 - 40 = \frac{34}{75} \times 100 $\)Simplify the right side:
$\( h_2 - 40 = \frac{34 \times 4}{3} $\) $\( h_2 - 40 = \frac{136}{3} $\)Add 40 to both sides:
$\( h_2 = 40 + \frac{136}{3} $\)Find a common denominator to add the terms:
$\( h_2 = \frac{40 \times 3}{3} + \frac{136}{3} = \frac{120}{3} + \frac{136}{3} $\) $\( h_2 = \frac{120 + 136}{3} = \frac{256}{3} $\)Approximating the Result
Convert the fraction to a decimal value:
$\( h_2 = \frac{256}{3} \approx 85.333... $\)The calculated height of the taller tower is approximately 85.33 feet.
Matching with Provided Options
The options given were:
- 85 feet
- 110 feet
- 125 feet
- 140 feet
The calculated value \(h_2 \approx 85.33\) feet is closest to the first option.
Conclusion
Based on the calculations derived from the geometric setup where the pole is located before the shorter tower, the height of the taller tower is approximately 85 feet.
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