document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192199>Question 1192199</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Variables x and y are such that, when e^y is plotted against x^2, a straight line graph passing through<BR>\n" );
document.write( "the points (0.2, 1) and (0.5, 1.6) is obtained.<BR>\n" );
document.write( "(i) Find the values of e^y when x = 0.<BR>\n" );
document.write( "(ii) Express y in terms of x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849072 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**(i) Find the values of e^y when x = 0:**\r<BR>\n" );
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document.write( "1. **Recognize the linear relationship:** Since plotting *eʸ* against *x²* gives a straight line, we can express this relationship as:\r<BR>\n" );
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document.write( "   *eʸ = mx² + c*\r<BR>\n" );
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document.write( "   where *m* is the slope and *c* is the y-intercept.\r<BR>\n" );
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document.write( "2. **Calculate the slope (m):**  We are given two points on the line: (x², eʸ) = (0.2², 1) and (0.5², 1.6).\r<BR>\n" );
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document.write( "   *m = (change in eʸ) / (change in x²)*<BR>\n" );
document.write( "   *m = (1.6 - 1) / (0.5² - 0.2²)*<BR>\n" );
document.write( "   *m = 0.6 / (0.25 - 0.04)*<BR>\n" );
document.write( "   *m = 0.6 / 0.21*<BR>\n" );
document.write( "   *m = 20/7*\r<BR>\n" );
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document.write( "3. **Calculate the y-intercept (c):** We can use either of the given points and the slope to find *c*. Let's use (0.2², 1):\r<BR>\n" );
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document.write( "   *1 = (20/7)(0.2²) + c*<BR>\n" );
document.write( "   *1 = (20/7)(0.04) + c*<BR>\n" );
document.write( "   *1 = 0.8/7 + c*<BR>\n" );
document.write( "   *c = 1 - 0.8/7*<BR>\n" );
document.write( "   *c = 6.2/7*\r<BR>\n" );
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document.write( "4. **Find *eʸ* when *x = 0*:** Substitute *x = 0* into the equation *eʸ = mx² + c*:\r<BR>\n" );
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document.write( "   *eʸ = (20/7)(0²) + 6.2/7*<BR>\n" );
document.write( "   *eʸ = 6.2/7*\r<BR>\n" );
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document.write( "   Therefore, when *x = 0*, *eʸ = 6.2/7* (approximately 0.886).\r<BR>\n" );
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document.write( "**(ii) Express y in terms of x:**\r<BR>\n" );
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document.write( "1. **We have the equation:** *eʸ = mx² + c*\r<BR>\n" );
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document.write( "2. **Substitute the values of *m* and *c*:**\r<BR>\n" );
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document.write( "   *eʸ = (20/7)x² + 6.2/7*\r<BR>\n" );
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document.write( "3. **Take the natural logarithm of both sides:**\r<BR>\n" );
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document.write( "   *ln(eʸ) = ln((20/7)x² + 6.2/7)*\r<BR>\n" );
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document.write( "4. **Simplify:**\r<BR>\n" );
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document.write( "   *y = ln((20/7)x² + 6.2/7)*\r<BR>\n" );
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document.write( "Therefore, *y = ln((20/7)x² + 6.2/7)*.<BR>\n" );
document.write( " </SPAN>\n" );
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