Please anyone solve these questions asap. I appreciate. Thanks Problem 7.16 Cons

Question

Please anyone solve these questions asap.
I appreciate.
Thanks

Problem 7.16 Consider the following variant of Buffon's needle problem from Example 7.4. A rectangular card with side lengths a and bis dropped at random on the floor. It is assumed that the length Va +b2 of the diagonal of the card is smaller than the distance D between the parallel lines on the floor. Show that the probability of the card intersecting one of the lines is given by 2(a--b Problem 7.17 choose randomly a point within a circle with radius r and construct the (unique) chord with the chosen point as its midpoint. What is the probability that the chord is longer than a side of an equilateral triangle inscribed in the circle? Problem 7.18 A stick is broken in two places. The breaking points are chosen at random on the stick, independently of each other. What is the probabil- ity that a triangle can be formed with the three pieces of the broken stick? Hint: the sum of the lengths of any two pieces must be greater than the length of the third piece. Problem 7.19 You choose a number v at random from (0,1) andnext a number w at random from (0,1 v). What is the probability that a triangle can be formed with the side lengths v, w and 1 w? Hint: represent was y(1

Explanation / Answer

Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint. The chord is longer than a side of the inscribed triangle if the chosen point falls within a concentric circle of radius 1/2 the radius of the larger circle. The area of the smaller circle is one fourth the area of the larger circle, therefore the probability a random chord is longer than a side of the inscribed triangle is 1/4.

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