document.write( "Question 832321: The manager of a furniture factory finds that it costs $2,200 to manufacture 100 chairs in one day and $4,800 to produce 300 chairs in one day.
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\n" ); document.write( "\n" ); document.write( "(b) what is the slope of the line in part (a) and what does it represent?\r
\n" ); document.write( "\n" ); document.write( "(c) what is the y- intercept of this line, and what does it represent? \n" ); document.write( "

Algebra.Com's Answer #502035 by math-vortex(648) $\"\"$ $\"About$

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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "The manager of a furniture factory finds that it costs $2,200 to manufacture 100 chairs in one day and $4,800 to produce 300 chairs in one day.\r\n" );
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document.write( "(a) Assuming that relationship between cost and the number of chairs produced is linear, find an equation that express this relationship.\r\n" );
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document.write( "(b) what is the slope of the line in part (a) and what does it represent?\r\n" );
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document.write( "(c) what is the y- intercept of this line, and what does it represent?\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "(a) Assume that relationship between cost and the number of chairs produced is linear. Find an equation that express this relationship.\r\n" );
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document.write( "This sentence tells us to find an equation relating the cost to the number of chairs. Since the relationship is linear, this will be the equation for a line.\r\n" );
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document.write( "First choose variables. I like n and C because they remind me of number and cost, but we could choose any letters, x and y, whatever.\r\n" );
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document.write( "Let n be the number of chairs produced.\r\n" );
document.write( "Let C be the cost (in dollars).\r\n" );
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document.write( "We need to determine which of these variable is independent and which in dependent. In this problem, \r\n" );
document.write( "C is the dependent variable because the Cost varies depending upon how many chairs are produced. This \r\n" );
document.write( "tells us the the horizontal axis is number of chairs produced n, and the vertical axis is the cost C.\r\n" );
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document.write( "In this problem we will follow the same process that we follow to come up with the equation for a line when we know two points. Don't let all the words here worry you too much.\r\n" );
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document.write( "First, we know that it costs $2,200 to produce 100 chairs. We can make an ordered pair out of this.\r\n" );
document.write( "(n, C) = (100, 2200)\r\n" );
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document.write( "Second, we know that it costs $4800 to produce 300 chairs. The order pair for this data is\r\n" );
document.write( "(n, C) = (300, 4800)\r\n" );
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document.write( "These points are on the line. We use them to find the slope m of the line.  Remember that the slope is the\r\n" );
document.write( "\"rise over the run,\" or the vertical change divided by the horizontal change.\r\n" );
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document.write( "To find vertical change, we subtract \"second C minus first C:\" 4800-2200 = 2600. From the first data point to the second, the cost increased by $2,600. \r\n" );
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document.write( "To find the horizontal change, we subtract \"second n minus first n:\" 300-100 = 200. From the first data \r\n" );
document.write( "point to the second, the number of chairs increased by 200.\r\n" );
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document.write( "The slope for our line is [vertical change] / [horizontal change] or \r\n" );
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document.write( "[change in Cost] / [change in number of chairs produced] = 2600/200\r\n" );
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document.write( "This is a fraction, so we can simplify it. Divided the numerator and denominator by 200\r\n" );
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document.write( "The slope for the line is 13/1, or 13. Recall that the numerator in this fraction is Cost (in dollars), and the \r\n" );
document.write( "denominator is the number of chairs produced. The slope tells that it costs $2600 to produced 200 \r\n" );
document.write( "chairs, but it also tells (after we reduced) that it costs $13 to produce 1 chair.\r\n" );
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document.write( "Now that we have the slope, we use one of our points and the slope to write an equation. The point-slope \r\n" );
document.write( "form for a linear equation is  where m is the slope and  and  are values for a point on the line. \r\n" );
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document.write( "We will use qs for the slope and  (100, 2200) for the point.\r\n" );
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document.write( "C - 2200 = 13(x - 100)\r\n" );
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document.write( "Simplify.\r\n" );
document.write( "C - 2200 = 13x - 1300\r\n" );
document.write( "C = 13x -1300 + 2200\r\n" );
document.write( "C = 13x + 900\r\n" );
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document.write( "This is the equation that expresses the relationship between the number of chairs produced and the cost.\r\n" );
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document.write( "(b) What is the slope of the line in part (a) and what does it represent?\r\n" );
document.write( "The slope of the line is 13 and it represents the COST PER CHAIR PRODUCED ($13 per chair).\r\n" );
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document.write( "(c) what is the y- intercept of this line, and what does it represent?\r\n" );
document.write( "Recall that a y-intercept is the place when the line crosses the y-axis, where x = 0. In this problem, the\r\n" );
document.write( "y-axis is the Cost axis, so\r\n" );
document.write( "The y-intercept is the C-intercept, the C-vaue when n = 0. Substitute 0 for n in the equation.\r\n" );
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document.write( "C = 13n + 900\r\n" );
document.write( "C = 13(0) + 900\r\n" );
document.write( "C = 900\r\n" );
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document.write( "This is the cost to produce no chairs. It is often refried to as the start-up cost.\r\n" );
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document.write( "Look at the equation C = 13n + 900\r\n" );
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document.write( "It tells you that cost to produce n chairs is $900 to get started, plus $13 for each chair you produce.\r\n" );
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document.write( "Hope this helps! Feel free to email if you have any questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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