Snow Big Deal
You are standing on a straight road in the middle of a snowy field. Your maximum walking speed is 4 mph on the road, and 2 mph in the field. Let A be the area, in square miles, of the region of the field you could reach within an hour. Find ⌊ 1 0 A ⌋ .
Details and assumptions
- The width of the road is negligible.
- You can be walking in any direction at any time.
- You don't need to be walking for the entire hour.
This problem is based on a tweet by James Tanton.)
The answer is 180.
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2 solutions
what would be the locus of the point of max. displacement from centre ?
I just wanted to say that I really liked this problem.
Suppose we start at the origin. Take the circle of radius 2 around the origin, and take the right triangle under the line x + y 3 = 4 , reflected through both axes to create a diamond. Then the region we're looking for is the union of those two regions. I get 4 π / 3 + 8 3 for its area A , and ⌊ 1 0 A ⌋ = 1 8 0 .
The proof is via some basic calculus that I'll just outline; basically, the time it takes to get to a point ( a , b ) in the first quadrant is 4 a + b 3 as long as a 3 < b , and b / 2 otherwise. In the former case, the correct path is to go a − b 3 miles on the road, then b 3 miles in the field, at an angle of 6 0 degrees to the road. In the latter case, the best path is a straight line through the field.
Details left to the reader.
I don't think you mean the union of those two regions, since the circle is contained entirely within the diamond. But it is a difficult shape to describe!
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Ah, good point. I want the diamond minus the part above and below the circle. (The circle is tangent to the diamond at the points ( ± 1 , ± 3 ) .)
please can u explain in simple terms....
Something is completely wrong somewhere. Even if the walking speed on the snow field is increased to 4 mph, the maximum perimeter that can be covered in 1 hour = 4 miles. The maximum area that can be enclosed by a given perimeter of 4 miles is when the enclosed region is a circle.
Perimeter = 2 π r = 4 ==> r = 2/π miles. Area, A = π (r^2) = π {(2/π)^2} = 4/π = 1.2732395 square miles. In this case, Floor(10*A) = 12
So, how can the answer be 180 when when walking speed on the snow field is halved to 2 mph in comparison with the above case.
The answer I get is = Floor{10*(1/π)} = 3.
Please, clarify.
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It sounds like you're talking about walking a path, returning back to your starting point by the end of an hour, and finding the area of the region you just enclosed. That is not what the problem is describing/asking.
Here's another way to ask it: You start from point X , and after an hour of walking, you are at point Y . The set of all possible points Y form a region. What is the area of that region?
This is not a solution, if I may say.
It is a little more than a disclosure of the results...
I should mention that the description of the path has the wrong dimensions: the correct path is to go a − b / 3 miles on the road, then 2 b / 3 miles in the field, at an angle of 6 0 degrees. The times are correct as stated.
This is about finding the envelope of circles starting with r = 2 at (0,0) to r = 0 at (4,0) and to r = 0 at (-4,0). It's simply a kite formed by 2 equilateral triangles of altitudes = 4, except for top and bottom rounded corners made by the tangent circle at (0,0) of radius 2. The exact area is 4(2√3 + π/3). No calculus is necessary for this one.