Two towers \(A\) and \(B\) of height 23 m and 11 m respectively, stand 9 m apart. A straight rod is joined to the two tops of the towers. A monkey sitting on the top of \(A\), climbs the rod to reach the top of \(B\). If the monkey takes 5 minutes to reach the other end, what is the average speed of the monkey?
This problem involves calculating the average speed of a monkey traveling between the tops of two towers of different heights using a connecting rod.
Towers Data Analysis
We are given the following information:
- Height of Tower A: \(h_A = 23 \text{ m}\)
- Height of Tower B: \(h_B = 11 \text{ m}\)
- Distance between the towers: \(d = 9 \text{ m}\)
- Time taken by the monkey to travel from top A to top B: \(T = 5 \text{ minutes}\)
Rod Distance Calculation
The monkey travels along the straight rod connecting the tops of the towers. The path taken by the monkey forms the hypotenuse of a right-angled triangle.
The vertical side of this triangle is the difference in the heights of the two towers:
Vertical difference: \( \Delta h = |h_A - h_B| = |23 \text{ m} - 11 \text{ m}| = 12 \text{ m} \)
The horizontal side of the triangle is the distance between the towers:
Horizontal distance: \( d = 9 \text{ m} \)
Using the Pythagorean theorem (\(a^2 + b^2 = c^2\)), we can find the length of the rod (the distance the monkey travels):
Let \( L \) be the length of the rod.
\( L^2 = (\Delta h)^2 + d^2 \)
\( L^2 = (12 \text{ m})^2 + (9 \text{ m})^2 \)
\( L^2 = 144 \text{ m}^2 + 81 \text{ m}^2 \)
\( L^2 = 225 \text{ m}^2 \)
\( L = \sqrt{225 \text{ m}^2} \)
\( L = 15 \text{ m} \)
So, the total distance the monkey travels is 15 meters.
Speed Calculation Steps
Average speed is defined as the total distance traveled divided by the total time taken.
\( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)
We have the distance \( L = 15 \text{ m} \) and the time \( T = 5 \text{ minutes} \).
Calculating the average speed in meters per minute:
\( v = \frac{15 \text{ m}}{5 \text{ min}} = 3 \text{ m/min} \)
Speed Unit Conversion
The options are given in cm/sec. We need to convert the calculated speed from m/min to cm/sec.
First, convert the distance from meters to centimeters:
\( 1 \text{ m} = 100 \text{ cm} \)
\( L = 15 \text{ m} = 15 \times 100 \text{ cm} = 1500 \text{ cm} \)
Next, convert the time from minutes to seconds:
\( 1 \text{ minute} = 60 \text{ seconds} \)
\( T = 5 \text{ minutes} = 5 \times 60 \text{ sec} = 300 \text{ sec} \)
Now, calculate the average speed in cm/sec:
\( v = \frac{1500 \text{ cm}}{300 \text{ sec}} \)
\( v = 5 \text{ cm/sec} \)
Therefore, the average speed of the monkey is 5 cm/sec.
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