answer:The total distance S traveled by the ball can be determined by summing up the distance fallen during each of the bounces. When the ball drops for the first time, it falls 32 meters. As it bounces back to half of the previous height each time, the distances are as follows: 1st fall: 32 meters, 1st bounce: 32/2 = 16 meters, 2nd fall: 16 meters (same as the 1st bounce), 2nd bounce: 16/2 = 8 meters, 3rd fall: 8 meters (same as the 2nd bounce), 3rd bounce: 8/2 = 4 meters, 4th fall: 4 meters (same as the 3rd bounce), 4th bounce: 4/2 = 2 meters, 5th fall: 2 meters (same as the 4th bounce), 5th bounce: 2/2 = 1 meter, and 6th fall: 1 meter (same as the 5th bounce). Note that after the sixth fall, the ball does not bounce back, so we do not double the final distance. Now, summing these distances up: S = 32 + (16 + 16) + (8 + 8) + (4 + 4) + (2 + 2) + (1 + 1) S = 32 + 2(16 + 8 + 4 + 2 + 1) S = 32 + 2(16 + 8 + 4 + 2 + 1) Using the formula for the sum of a geometric series S_n = a(1 - r^n)/(1 - r) where a is the first term and r is the common ratio, here a = 16 and r = 1/2, and we sum the first five terms, S = 32 + 2 times 16 times left( frac{1 - left(frac{1}{2}right)^{5}}{1 - frac{1}{2}} right) S = 32 + 32 times left( 1 - frac{1}{32} right) S = 32 + 32 - 1 S = 63 Therefore, the total distance S traveled by the ball by the time it hits the ground for the sixth time is boxed{63} meters.

answer:1. Consider the right triangle ( triangle ABC ) with a ( 30^circ ) angle at vertex ( A ). Without loss of generality, we assume ( AC = 1 ). Therefore, the length of the other leg, ( BC ), can be determined using the properties of a ( 30^circ-60^circ-90^circ ) triangle. Specifically, ( BC = sqrt{3} ). 2. Calculate the area of ( triangle ABC ): [ text{Area of } triangle ABC = frac{1}{2} times AC times BC = frac{1}{2} times 1 times sqrt{3} = frac{sqrt{3}}{2} ] 3. To form a square with the same area as ( triangle ABC ), determine the side length of the square: [ text{Side length of the square} = sqrt{frac{sqrt{3}}{2}} ] 4. The goal is to cut the ( triangle ABC ) into four parts that can be rearranged to form a square with side length ( sqrt{frac{sqrt{3}}{2}} ). 5. The first step in this process is to form a rectangle from ( triangle ABC ). Cut along the midline of ( triangle ABC ) from ( A_1 ) (midpoint of ( AB )) to ( C_1 ) (midpoint of ( AC )), and rotate the triangle part ( triangle A_1 C_1 B ) around ( C_1 ) by ( 180^circ ) to form a rectangle. The resulting rectangle ( A_1 A_1' A C ) will have sides ( A_1 A_1' = AC = 1 ) and ( A_1' A = CA_1 = frac{sqrt{3}}{2} ).  6. Next, find a point ( P ) on side ( CA ) such that ( CP = sqrt{frac{sqrt{3}}{2}} ). Similarly, locate point ( Q ) on side ( A_1 A_1' ) at the same distance from ( A_1' ). 7. Cut along the line ( A_1 P ). Translate the cut-off triangle ( triangle A_1 P C ) parallel to ( A_1 P ) such that its new position aligns with ( Q ). The new configuration ensures that the piece slides along to meet ( Q ), forming the quadrilateral ( Q A_1' P^* C^* ), which should be a square with the side length ( sqrt{frac{sqrt{3}}{2}} ).  8. Ensure the alignment and arrangement correctly form the desired square. Each cutting and rotation step aligns with the provided measurements to preserve both shape integrity and area equality. 9. Upon validating the cutting steps and transformations, the final configuration consists of four parts reassembled to form a square with the desired properties as indicated.  # Conclusion: By following the described steps, we successfully transformed ( triangle ABC ) into four parts that can be rearranged into a square with the side length ( sqrt{frac{sqrt{3}}{2}} ). blacksquare

answer:First, we simplify the given angle frac{17 pi}{4} by subtracting 4 pi (which is equivalent to a full circle) to bring the angle within the range of [0, 2pi). frac{17 pi}{4} - 4 pi = frac{17 pi - 16 pi}{4} = frac{pi}{4} Now, we can find the value of sin frac{pi}{4}: sin frac{pi}{4} = frac{sqrt{2}}{2} Therefore, the correct answer is option A: boxed{frac{sqrt{2}}{2}}.

answer:Let's denote the original purchase price of the product as P. Bill made a profit of 10% on the original purchase price, so his original selling price (S) would be: S = P + 0.10P S = 1.10P If Bill had purchased the product for 10% less, the purchase price would have been: New purchase price = P - 0.10P New purchase price = 0.90P If he sold it at a profit of 30%, the new selling price (S_new) would be: S_new = 0.90P + 0.30(0.90P) S_new = 0.90P + 0.27P S_new = 1.17P According to the problem, the new selling price is 28 more than the original selling price: S_new = S + 28 1.17P = 1.10P + 28 Now, let's solve for P: 1.17P - 1.10P = 28 0.07P = 28 P = 28 / 0.07 P = 400 Now that we have the original purchase price (P), we can find the original selling price (S): S = 1.10P S = 1.10 * 400 S = 440 So, the original selling price was boxed{440} .