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12 May 2021, 04:45
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B
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Difficulty:
35%
(medium)
Question Stats:
73% (02:10) correct
27%
(02:02)
wrong
based on 26
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Noisy the dog walks from his doghouse at A, to his food bowl at B. Usually he runs the 24 meter distance in a straight line, but on Tuesday there was a little puddle in his way, and he decided to go around it by running along a circular path centered at P, as shown in the figure above. If the radius of the circular bypass is 4, what is the distance Noisy ran when going from A to B?
A. \(16+6π\)
B. \(24−4√2+6π\)
C. \(16+8π\)
D. \(24−4√2+8π\)
E. \(24+6π\)
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16 May 2021, 10:26
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To get the new distance, we must calculate:
The original distance (24 meters) - The unused distance from A' C' B' + The new distance from A' C B'
The unused distance from A' C' B' is similar to a right triangle.
Since we know that A' P B' is 90 degree, and we know that the radius of semi-circle P is 4 meters, then by the pythagoran theorem of (45, 45, 90), we know that the distance of A' C' B' is 4\sqrt{2}
The new distance from A' C B' is basically 3/4 of the perimeter of the circle P, since it covers 270/360 degrees of the circle.
Therefore the distance A' C B' is pi*2*r*(270/360) = pi*2*(4 meters)*(3/4) = 6pi
Substituting the calculation back to the main equation results in:
24 meters - 4\sqrt{2} meters + 6pi meters
which is the option B
The original distance (24 meters) - The unused distance from A' C' B' + The new distance from A' C B'
The unused distance from A' C' B' is similar to a right triangle.
Since we know that A' P B' is 90 degree, and we know that the radius of semi-circle P is 4 meters, then by the pythagoran theorem of (45, 45, 90), we know that the distance of A' C' B' is 4\sqrt{2}
The new distance from A' C B' is basically 3/4 of the perimeter of the circle P, since it covers 270/360 degrees of the circle.
Therefore the distance A' C B' is pi*2*r*(270/360) = pi*2*(4 meters)*(3/4) = 6pi
Substituting the calculation back to the main equation results in:
24 meters - 4\sqrt{2} meters + 6pi meters
which is the option B
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16 May 2021, 22:58
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Total straight line distance from A to B = 24
Noisy follows the circular path (Circle: A to start of circle + end of the circle to B)
If we connect the Start and endpoint of the circle we will get the Hypotenuse with the other two sides(radius = 4) and angle P = \(90^o\).
Hence, hypotenuse will be \(4\sqrt{2}\) - This distance is not covered.
Distance now is 24 - \(4\sqrt{2}\).
Circular path: Length of bigger arc = Circumference of the circle- Length of minor arc
=> \(2πr - \frac{90}{360} * 2πr\)
=> \(8π - \frac{1}{4} * 2 * π * 4\)
=> 6π
Distance covered: 24 - \(4\sqrt{2}\) + 6π
Answer B
Noisy follows the circular path (Circle: A to start of circle + end of the circle to B)
If we connect the Start and endpoint of the circle we will get the Hypotenuse with the other two sides(radius = 4) and angle P = \(90^o\).
Hence, hypotenuse will be \(4\sqrt{2}\) - This distance is not covered.
Distance now is 24 - \(4\sqrt{2}\).
Circular path: Length of bigger arc = Circumference of the circle- Length of minor arc
=> \(2πr - \frac{90}{360} * 2πr\)
=> \(8π - \frac{1}{4} * 2 * π * 4\)
=> 6π
Distance covered: 24 - \(4\sqrt{2}\) + 6π
Answer B
