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An infinite plane can be tiled using identical triangles, quadrilaterals, or hexagons with no overlaps and no gaps.
Is it possible to tile an infinite plane using identical pentagons?
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A solid disk, a solid sphere, and a hoop of identical mass m and radius R are mounted at rest on frictionless axles. Each object has a string wrapped around its circumference, and the strings are each pulled with an identical force F for the same time interval; the strings do not slip.
Which object will have the greatest rotational kinetic energy at the end of the time interval?
Note: The mass of the spokes on the hoop is negligible.
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Lines are drawn through point P parallel to the sides of triangle ABC. The resulting 3 triangles have areas 4, 9, and 49.
What is the area of triangle ABC?
The areas of these 3 smaller triangles are 4, 9, and 49. What's the area of the big, outer triangle?
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In this rectangular window frame, a spider begins to weave its web. The basic framework consists of three threads, which start from points A,B, and C and are glued together at point D.
What is the length lA of the first thread in centimeters?
Assumptions: Without tension, all threads have the same length, but they are stretched many times their original length. Thus, the tension force Fi≈−kli acting along a thread is approximately proportional to its total length li. While the lengths li of the threads can be different, the spring constant k is the same for all threads. Apart from the tension, no further forces act on the spider threads.
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a,b, and x are real numbers satisfying the following system of equations: a+bab==3x−24x2−3x−4. What is the minimum value of a2+b2 (to three decimal places)?
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