document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170427>Question 1170427</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 𝐏𝐫𝐨𝐛𝐥𝐞𝐦 𝟒    <BR>\n" );
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document.write( "(b)  During a nationwide program to immunize the population against a new strain of the flu,  public health officials determined that the cost of inoculating x% of the susceptible population <BR>\n" );
document.write( "1.85x<BR>\n" );
document.write( "would be approximately   C(x) =   million dollars.   x<BR>\n" );
document.write( "(i)	What would it cost to providing immunization to the first 20% of the susceptible  population?                                          <BR>\n" );
document.write( "(ii)	What would it cost to providing immunization to the next 30% of the susceptible population?                                         <BR>\n" );
document.write( "(iii)	Suppose 17 million dollars are available for providing immunization. What percentage  of the susceptible population will not receive immunization?                   <BR>\n" );
document.write( "(iv)	If money was not a problem will they be able to providing immunization to the entire  susceptible  population?                                     <BR>\n" );
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document.write( "(c)  Determine the values of x for which the function f   <BR>\n" );
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document.write( "is  discontinuous.     <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851188 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=851188\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each part of the problem.\r<BR>\n" );
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document.write( "**(b) Immunization Costs**\r<BR>\n" );
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document.write( "We are given the cost function:<BR>\n" );
document.write( "$$C(x) = \frac{1.85x}{100-x}$$<BR>\n" );
document.write( "where $x$ is the percentage of the susceptible population immunized, and $C(x)$ is in million dollars.\r<BR>\n" );
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document.write( "**(i) Cost for the first 20%:**\r<BR>\n" );
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document.write( "We need to find $C(20)$.<BR>\n" );
document.write( "$$C(20) = \frac{1.85(20)}{100-20} = \frac{37}{80} = 0.4625$$<BR>\n" );
document.write( "So, it would cost 0.4625 million dollars, or $462,500.\r<BR>\n" );
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document.write( "**(ii) Cost for the next 30%:**\r<BR>\n" );
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document.write( "If we immunize the first 20%, then we need to immunize from 20% to 50%.<BR>\n" );
document.write( "First, find the cost of immunizing 50%:<BR>\n" );
document.write( "$$C(50) = \frac{1.85(50)}{100-50} = \frac{92.5}{50} = 1.85$$<BR>\n" );
document.write( "The cost for the next 30% is $C(50) - C(20)$:<BR>\n" );
document.write( "$$1.85 - 0.4625 = 1.3875$$<BR>\n" );
document.write( "So, it would cost 1.3875 million dollars, or $1,387,500.\r<BR>\n" );
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document.write( "**(iii) Percentage not immunized with $17 million:**\r<BR>\n" );
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document.write( "We need to find $x$ such that $C(x) = 17$:<BR>\n" );
document.write( "$$17 = \frac{1.85x}{100-x}$$<BR>\n" );
document.write( "$$17(100-x) = 1.85x$$<BR>\n" );
document.write( "$$1700 - 17x = 1.85x$$<BR>\n" );
document.write( "$$1700 = 18.85x$$<BR>\n" );
document.write( "$$x = \frac{1700}{18.85} \approx 90.1856764$$<BR>\n" );
document.write( "So, approximately 90.19% of the population can be immunized.<BR>\n" );
document.write( "The percentage not immunized is $100 - x$:<BR>\n" );
document.write( "$$100 - 90.19 \approx 9.81$$<BR>\n" );
document.write( "Approximately 9.81% of the population will not receive immunization.\r<BR>\n" );
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document.write( "**(iv) Immunization for the entire population:**\r<BR>\n" );
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document.write( "If we want to immunize 100% of the population, we need to find $C(100)$:<BR>\n" );
document.write( "$$C(100) = \frac{1.85(100)}{100-100} = \frac{185}{0}$$<BR>\n" );
document.write( "This is undefined, as we cannot divide by zero.<BR>\n" );
document.write( "Therefore, it is impossible to immunize the entire susceptible population.\r<BR>\n" );
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document.write( "**(c) Discontinuity of f(x)**\r<BR>\n" );
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document.write( "You didn't provide the function f(x). However, I can explain how to find the discontinuities of a function.\r<BR>\n" );
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document.write( "**Finding Discontinuities:**\r<BR>\n" );
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document.write( "* **Rational Functions:** If $f(x)$ is a rational function (a fraction where the numerator and denominator are polynomials), then the function is discontinuous where the denominator is zero.<BR>\n" );
document.write( "* **Piecewise Functions:** If $f(x)$ is a piecewise function, then the function is discontinuous where the pieces do not meet.<BR>\n" );
document.write( "* **Other Functions:** If $f(x)$ involves other functions like logarithms or square roots, then the function is discontinuous where those functions are undefined.\r<BR>\n" );
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document.write( "**Example:**\r<BR>\n" );
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document.write( "Let's say $f(x) = \frac{x+1}{x-2}$.\r<BR>\n" );
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document.write( "To find the discontinuities, we set the denominator to zero:<BR>\n" );
document.write( "$x - 2 = 0$<BR>\n" );
document.write( "$x = 2$\r<BR>\n" );
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document.write( "Therefore, $f(x)$ is discontinuous at $x = 2$.\r<BR>\n" );
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document.write( "**Please provide the function f(x) so I can determine its discontinuities.**<BR>\n" );
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