A tight fitting band is wrapped around the Equator. Another circular band whose length is 15 m more lies at a certain height over the first band. A group of human beings attempt to pass under the longer band. Can they walk under it? (Earth’s circumference is roughly 40000 km. The height of human beings is between 1 & 2m)
Yes
This problem involves understanding the relationship between the circumference and radius of a circle and how a small increase in circumference affects the radius, regardless of the initial size of the circle.
Equator Band Circumference
Let the circumference of the tight-fitting band around the Equator be \(C_1\). If the radius of the Earth (and thus this band) is \(r_1\), then the circumference is given by the formula:
| \(C_1 = 2\pi r_1\) |
The second circular band is 15 meters longer than the first. Let its circumference be \(C_2\) and its radius be \(r_2\).
| \(C_2 = C_1 + 15\) meters |
| \(C_2 = 2\pi r_2\) |
Height Difference Calculation
We want to find the height between the two bands. Assuming the bands are concentric circles, this height is the difference between their radii, \(h = r_2 - r_1\). We can find this difference by substituting the first equation into the second:
| \(2\pi r_2 = (2\pi r_1) + 15\) |
Now, let's solve for the difference \(r_2 - r_1\):
| Divide both sides by \(2\pi\): |
| \(r_2 = r_1 + \frac{15}{2\pi}\) |
| Rearrange the terms to find the difference: |
| \(r_2 - r_1 = \frac{15}{2\pi}\) |
The height \(h\) of the second band above the first band is \(h = r_2 - r_1\). Let's calculate its approximate value using \(\pi \approx 3.14159\):
| \(h = \frac{15}{2 \times \pi}\) meters |
| \(h \approx \frac{15}{2 \times 3.14159}\) meters |
| \(h \approx \frac{15}{6.28318}\) meters |
| \(h \approx 2.387\) meters |
Can Humans Walk Under the Band?
The calculated height difference between the two bands is approximately 2.387 meters. The height of human beings is stated to be between 1 and 2 meters. Since the height of the gap under the longer band (approx 2.387 m) is greater than the maximum height of a human being (2 m), a group of human beings can indeed walk under the longer band.
The large circumference of the Earth (40000 km) is irrelevant to the calculation of the height difference; only the *difference* in circumference matters.
Therefore, the answer is Yes, they can walk under the longer band.
Important Questions from Problem Solving
Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 - Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
(84, 73, 157) (96, 88, 184)
- Arrange the problem-solving strategy.
(A) Generate many possible solutions
(B) Determine how to implement the solution and evaluate the success of the solution
(C) Make a decision
(D) Evaluate each solution
(E) Define the problem
Choose the correct answer from the options given below. - A cyclist starting from point $X$ travels straight towards east for 3km and then turns right and covers 4km and reaches point $Y$. What is the shortest distance between point $X$ and $Y$?
- Read the given paragraph and answer the question.
There are five persons A, G, M, J and D. Every person has to select one item each from a box containing items numbered from 1 to 5. A and J selected items 5 and 2 respectively. G cannot select item 3, while M cannot select item 3 or 5.
Which item can G select? - Read the given information carefully and answer the question that follows.
- There are four persons: P, Q, R and S.
- One of them is a doctor and plays football and cricket.
- P and Q are teachers.
- P plays badminton.
- The lawyer and both the teachers are swimmers.
- One of them also plays tennis.
- S plays carrom and is a lawyer.
- All the four persons play two games each and follow one profession.
