document.write( "Question 1198182: A cart weighing 40 lb is placed on a ramp inclined at 15° to the horizontal. The cart is held in place by a rope inclined at 60° to the horizontal, as shown in the figure. Find the force that the rope must exert on the cart to keep it from rolling down the ramp \n" ); document.write( "
Algebra.Com's Answer #831729 by ikleyn(52847)\"\" \"About 
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\n" ); document.write( "A cart weighing 40 lb is placed on a ramp inclined at 15° to the horizontal.
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\n" ); document.write( "\n" ); document.write( "            There is no any figure attached to the post.\r
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document.write( "The force that the rope exerts on the cart is called \"rope tension\", or simply \"tension\".\r\n" );
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document.write( "Let F be the tension.\r\n" );
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document.write( "Then it takes off  F*sin(60°) from the weight  W  of the cart, leaving only  W-F*sin(60°) as \r\n" );
document.write( "the net vertical force acting on the cart.\r\n" );
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document.write( "This vertical force  W-F*sin(60°)  can be decomposed by the usual way into the sum \r\n" );
document.write( "of two vectors/(components), one parallel to the incline and the other normal to it.\r\n" );
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document.write( "The component directed along the incline is  (W-F*sin(60°))*sin(15°).\r\n" );
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document.write( "This force/(component) is balanced by the component  F*cos(60°- 15°) = F*cos(45°)  directed along \r\n" );
document.write( "the incline  in the opposite direction.\r\n" );
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document.write( "Thus the equilibrium equation is\r\n" );
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document.write( "        (W-F*sin(60°) )*sin(15°) = F*cos(45°).\r\n" );
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document.write( "which gives\r\n" );
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document.write( "        F = \"%28W%2Asin%2815%5Eo%29%29%2F%28sin%2860%5Eo%29%2Asin%2815%5Eo%29+%2B+cos%2845%5Eo%29%29\".\r\n" );
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document.write( "Further,  sin(15°) = 0.2588;\r\n" );
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document.write( "        sin(60°)*sin(15°) + cos(45°) = \"%28sqrt%283%29%2F2%290.2588\" + \"sqrt%282%29%2F2%29\" = 0.9312,\r\n" );
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document.write( "        so  F = \"%2840%2A0.2588%29%2F0.9312\" = 11.12 pounds.\r\n" );
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document.write( "Answer.  The tension of the rope is 11.12 pounds.\r\n" );
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