[Beam freely supported at both ends, loaded uniformly. Origin at lowest point.] (c) . y = k(6l2x2 - 4lx3 + x1). [Beam embedded at one end only, loaded uniformly. Origin at fixed end.] 26. Find the altitude of a cylinder inscribed in a cone when the volume of the cylinder is a maximum. Ans. 3x = h. . 27. Find the altitude h of the greatest cylinder that can be cut out of a given sphere of radius R. Ans. h = 2R/√3. 28. Find the altitude h of the greatest cone that can be inscribed in a sphere of radius R. Ans. h 4R/3. 29. Find the altitude of a cone inscribed in a sphere which shall make the convex surface of the cone a maximum. Ans. R. 30. The velocity of waves of length x in deep water is proportional to √(x/a) + (a/x), where a is a certain linear magnitude. Show that the velocity is a minimum when x = a. 31. A weight of 1000 lbs. hanging 2 feet from one end of a lever is to be raised by an upward force at the other end. Supposing the lever to weigh 10 lbs. per foot, find its length that the force may be a minimum. Ans. 20 ft. 32. Find the most economical proportions for a covered box whose base is a rectangle with one side twice the other. 33. Find the most economical proportions for a conical tent of given capacity. r√2. Ans. h = r 0 123 456 0000 3010 4771 6021 6990 7782 8451 9031 9542 0000 0414 0792 1139 1461 1761 2041 2304 2553 2788 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 6435 6532 6628 6721 6812 6902 123456789 6021 6128 6232 6335 6990 7076 7160 7243 7782 7853 7924 7993 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 4 5 11 0414 0453 0492 0531 0569 0607 0645 0682 0792 0828 0864 0899 0934 0969 1004 1038 1139 1173 1206 1239 1271 1303 1335 1367 10 7898123 1234∞α 22 21 1 2.2 2.1 4.4 4.2 6.6 6.3 8.8 8.4 11.0 10.5 13.2 12.6 15.4 14.7 17.6 16.8 9 19.8 18.9 6 12.0 11.4 0719 0755 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 1234∞ a 9 18.0 17.1 5 9.0 8.5 6 10.8 10.2 7 12.6 11.9 8 14.4 13.6 9 16.2 15.3 16 15 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 4150 4166 4183 4200 4216 4232 4249 4265 4281 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 9 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 11.2 10.4 9 12.6 11.7 37 1234 GT 12 11 12 1.1 2.4 2.2 3.6 3.3 6.4 6.0 5 8.0 7.5 4298 6 9.6 9.0 4314 4330 4346 4362 4378 4393 4409 4425 4440 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 9 14.4 13.5 12345678 14 13 1.4 1.3 2.8 2.6 4.2 3.9 5.6 5.2 1234∞a 1.6 1.5 2 7 11.2 10.5 4456 8 12.8 12.0 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 4.8 4.4 5 6.0 55 6 7.2 6.6 7 8.4 7.7 8 9.6 8.8 10.8 9.9 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 9 8.1 7.2 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 72757284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 61 7853 78607868 | 7875 7882 7889 78967903 7910 7917 8116 8122 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1234 17 0 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 95629566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 6 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 9 5.4 123456789 123456789 5 0.5 1.0 15 2.0 2.5 3.0 3.5 4.0 4.5 4 04 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 |