document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139829>Question 1139829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A new car costs $21,000 depreciates to 75% of its value in 4 years. Round all answers to 2 decimal places, where necessary.<BR>\n" );
document.write( "a). Assume that the depreciation is linear. What is the linear function that models the value of this car ,t,<BR>\n" );
document.write( "years after purchase? <BR>\n" );
document.write( "b). Assume that the value of the car is given by an exponential function Ae^kt where A is the initial price of the car. Find the value of the constant k. Answer exactly. <BR>\n" );
document.write( "c). For the linear model used in part (a), find the value of the car 6 years after the purchase, then do the same for the exponential model used in part (b).Round your answer to 2 decimal places.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760329 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=760329\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> in the straight line depreciation model, you use the equation with the general form of y = mx + b, where m is the slope and b is the y-intercept.\r<BR>\n" );
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document.write( "y is the value of the car.<BR>\n" );
document.write( "x is the number of years from when the car is new.\r<BR>\n" );
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document.write( "the y-intercept is the value of the car when x = 0.<BR>\n" );
document.write( "this is when the car is new.<BR>\n" );
document.write( "the y-intercept is therefore 21000.\r<BR>\n" );
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document.write( "m is the average change in the value of the car each year.<BR>\n" );
document.write( "if the value of the car drops from 21000 to 3/4 * 21000 in 4 years, then the slope is (15750 - 21000) / 4 = -1312.5\r<BR>\n" );
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document.write( "the equation for straight line depreciation becomes y = -1312.5 + 21000.\r<BR>\n" );
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document.write( "the graph of that equation is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042501.jpg\"alt=\"$$$\" >\r<BR>\n" );
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document.write( "when the value of the car is given by the exponential formula of A * e^(kt), you would create the formula as follows.\r<BR>\n" );
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document.write( "the general form of the exponential form, as shown, is V = A * e^(kt)\r<BR>\n" );
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document.write( "V is the value of the car after t years have elapsed.<BR>\n" );
document.write( "A is the initial value of the car.<BR>\n" );
document.write( "e is the scientific constant of 2.718281828.....<BR>\n" );
document.write( "t is the number of years.<BR>\n" );
document.write( "k will be the interest rate per year.\r<BR>\n" );
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document.write( "A is equal to 21000.<BR>\n" );
document.write( "when t is 4, V is equal to 15750 (3/4 of the initial value of the car).\r<BR>\n" );
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document.write( "the formula becomes 15750 = 21000 * e^(4k).<BR>\n" );
document.write( "divide both sides of this formula to get 3/4 = e^(4k)<BR>\n" );
document.write( "take the natural log of both sides of this formula to get ln(3/4) = ln(e^4k).<BR>\n" );
document.write( "since ln(e^4k) = 4k * ln(e), and since ln(e) is equal to 1, the formula becomes ln(3/4) = 4 * k<BR>\n" );
document.write( "divide both sides of this formula by 4 to get ln(3/4) / 4 = k<BR>\n" );
document.write( "solve for k to get k = -.0719205181.\r<BR>\n" );
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document.write( "the formula becomes V = 21000 * e^(-.0719205181 * t).\r<BR>\n" );
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document.write( "to graph this formula,let y = V and let x = t.\r<BR>\n" );
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document.write( "the formula becomes y = 21000 * e^(-.0719205181 * x).\r<BR>\n" );
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document.write( "the graph of that equation is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042502.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "you in see that, in both graphs, when the value of x = 4, the value of the car is 15750.\r<BR>\n" );
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document.write( "the graph of the straight line depreciation is a straight line.\r<BR>\n" );
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document.write( "the graph of the exponential depreciation is a curve.\r<BR>\n" );
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document.write( "the value of the car drops the same amount each year with straight line depreciation.\r<BR>\n" );
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document.write( "the value of the car drops by the same percentage each year with exponential depreciation.\r<BR>\n" );
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document.write( "the percentage it drops each year is derived from the exponential formula.\r<BR>\n" );
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document.write( "the formula is V = A * e^(kt).\r<BR>\n" );
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document.write( "k is equal to -.0719205181.\r<BR>\n" );
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document.write( "when t = 1, e^(k * t) becomes e^(-.0719205181) which is equal to .9306048591.\r<BR>\n" );
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document.write( "each year, the value of the car is .9306048591 * the value of the car in the preceding year.\r<BR>\n" );
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document.write( "that's a drop in value of 1 - .9306048591 = .0693951409.\r<BR>\n" );
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document.write( "you can confirm this is true by looking at each year.\r<BR>\n" );
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document.write( "at year 0,the value is 21000<BR>\n" );
document.write( "at year 1,the value is .93... * 21000 = 19542.70...<BR>\n" );
document.write( "at year 2, the value is .93... * 19542.70... = 18186.53...<BR>\n" );
document.write( "at year 3, the value is .93... * 18186.53... = 16924.47...<BR>\n" );
document.write( "at year 4, the value is .93... * 16924.47... = 15750.\r<BR>\n" );
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document.write( "it will be exactly 15750 after 4 years because 4 years is how the value of k was determined.\r<BR>\n" );
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document.write( "looking at the two graphs, you will see that the value of the car becomes equal to 0 in 16 years with straight line depreciation, while the value of the car is equal to 6644.53... after 16 years with exponential depreciation.\r<BR>\n" );
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document.write( "this is because the value of the car drops less and less each year with exponential depreciation.\r<BR>\n" );
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document.write( "any questions, write to dtheophilis@gmail.com </SPAN>\n" );
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