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
Evaluate 5793405 x 99999
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
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