document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198078>Question 1198078</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You must use the constant product princivle to solve these problems. Answer al questions in the space<BR>\n" );
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document.write( "Suppose you drive the same route to work every day, Each morning you leave home at 8:20<BR>\n" );
document.write( "and arrive at work at 9:00. However, today you are late and do not leave until 8:30.\r<BR>\n" );
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document.write( "a.What IS the fractional decrease in time So that you will arrive at 9:00?<BR>\n" );
document.write( "b. by What fraction must your speed change so that you'll to get to work at 9:00?\r<BR>\n" );
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document.write( "2. Each time I stop at the gas station I put exactly $60 worth of gas into my car<BR>\n" );
document.write( "a. This week the gas prices have risen by 5%. Does this mean that I will get 5% less gas for<BR>\n" );
document.write( "my $60? Explain.\r<BR>\n" );
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document.write( "b. How much gas Will I get?\r<BR>\n" );
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document.write( "c. Suppose gas prices rise by a fraction r. What is the effect on the amount of gas my $60 will buy?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848341 by </B><A HREF=/tutors/aboutme.mpl?userid=onyulee><B>onyulee(41)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=onyulee><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ### Problem 1: Driving to Work\r<BR>\n" );
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document.write( "#### Part a: Fractional Decrease in Time<BR>\n" );
document.write( "- Original time: \( 40 \, \text{minutes} \) (from 8:20 to 9:00).<BR>\n" );
document.write( "- New time: \( 30 \, \text{minutes} \) (from 8:30 to 9:00).<BR>\n" );
document.write( "- Fractional decrease in time is given by:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\text{Fractional Decrease} = \frac{\text{Original Time} - \text{New Time}}{\text{Original Time}} = \frac{40 - 30}{40} = \frac{10}{40} = 0.25<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Thus, the fractional decrease in time is \( 0.25 \) or \( 25\% \).\r<BR>\n" );
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document.write( "#### Part b: Fractional Increase in Speed<BR>\n" );
document.write( "Using the constant product principle (\( \text{Speed} \times \text{Time} = \text{Distance} \)):<BR>\n" );
document.write( "- Original speed: \( v \), original time: \( 40 \, \text{minutes} \).<BR>\n" );
document.write( "- New speed: \( v_{\text{new}} \), new time: \( 30 \, \text{minutes} \).<BR>\n" );
document.write( "- Since the distance is the same:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "v \times 40 = v_{\text{new}} \times 30<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "v_{\text{new}} = \frac{40}{30} v = \frac{4}{3} v<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Fractional increase in speed is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\text{Fractional Increase} = \frac{v_{\text{new}} - v}{v} = \frac{\frac{4}{3}v - v}{v} = \frac{4}{3} - 1 = \frac{1}{3}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Thus, the fractional increase in speed is \( \frac{1}{3} \) or \( 33.33\% \).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Problem 2: Gas Station Purchase\r<BR>\n" );
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document.write( "#### Part a: Does a 5% Price Increase Reduce Gas by 5%?<BR>\n" );
document.write( "No. The relationship between price per gallon and the quantity of gas purchased is inverse. A 5% increase in price does not correspond to a 5% decrease in gas but will decrease the gas by slightly less.\r<BR>\n" );
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document.write( "#### Part b: How Much Gas Will I Get?<BR>\n" );
document.write( "Let:<BR>\n" );
document.write( "- Original price per gallon: \( p \).<BR>\n" );
document.write( "- New price per gallon: \( p_{\text{new}} = 1.05p \).<BR>\n" );
document.write( "- Gas obtained: \( g \), where \( g = \frac{60}{p_{\text{new}}} = \frac{60}{1.05p} = \frac{60}{p} \times \frac{1}{1.05} \).<BR>\n" );
document.write( "- Original gas purchased: \( g_{\text{original}} = \frac{60}{p} \).\r<BR>\n" );
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document.write( "The new amount of gas is reduced by a factor of \( \frac{1}{1.05} \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "g = g_{\text{original}} \times \frac{1}{1.05}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Part c: Effect of Fractional Price Increase \( r \)<BR>\n" );
document.write( "If gas prices increase by a fraction \( r \), the new price is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "p_{\text{new}} = (1 + r)p<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "The amount of gas purchased becomes:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "g = \frac{60}{p_{\text{new}}} = \frac{60}{(1 + r)p} = \frac{g_{\text{original}}}{1 + r}<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Thus, the amount of gas decreases inversely to \( 1 + r \). For every fraction \( r \) increase in price, the quantity of gas decreases by a factor of \( \frac{1}{1 + r} \). </SPAN>\n" );
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