document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170005>Question 1170005</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ship-Cleaning Robots<BR>\n" );
document.write( "When a ship arrives in port, an underwater robot cleans the outside of the hull in order to remove sea<BR>\n" );
document.write( "creatures, seaweed and dirt. The robot has magnetic wheels that allow it to crawl over the ship’s steel<BR>\n" );
document.write( "hull. Making the hull smooth in this way can save about 8% of fuel on the next voyage. It is important<BR>\n" );
document.write( "that the cleaning can be completed before the ship leaves port.<BR>\n" );
document.write( "(a) (3 marks) The amount of time taken to offload and reload cargo is 22.7 hours. Robot (Type A) cleans<BR>\n" );
document.write( "one side of the ship followed by the other side. The cleaning time depends on the amount of dirt on the<BR>\n" );
document.write( "hull and is Normally distributed with a mean of 9.3 hours and a standard deviation of 1.6 hours for one<BR>\n" );
document.write( "side of the ship. The time for the second side of the ship is also Normally distributed with a mean of 9.3<BR>\n" );
document.write( "hours and a standard deviation of 1.6 hours and is correlated with the time taken for the first side with a<BR>\n" );
document.write( "correlation coefficient of 0.85. What is the probability that the robot will have finished cleaning the<BR>\n" );
document.write( "ship’s hull before the cargo is offloaded and reloaded?<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 #851270 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=851270\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**(a) Probability of Finishing Cleaning Before Cargo Handling:**\r<BR>\n" );
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document.write( "1.  **Define Variables:**<BR>\n" );
document.write( "    * Let X be the cleaning time for the first side of the ship.<BR>\n" );
document.write( "    * Let Y be the cleaning time for the second side of the ship.<BR>\n" );
document.write( "    * X ~ N(μx = 9.3, σx = 1.6)<BR>\n" );
document.write( "    * Y ~ N(μy = 9.3, σy = 1.6)<BR>\n" );
document.write( "    * Correlation coefficient (ρ) = 0.85<BR>\n" );
document.write( "    * Cargo handling time = 22.7 hours\r<BR>\n" );
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document.write( "2.  **Total Cleaning Time (T):**<BR>\n" );
document.write( "    * T = X + Y\r<BR>\n" );
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document.write( "3.  **Mean of Total Cleaning Time (μt):**<BR>\n" );
document.write( "    * μt = μx + μy = 9.3 + 9.3 = 18.6 hours\r<BR>\n" );
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document.write( "4.  **Variance of Total Cleaning Time (σt²):**<BR>\n" );
document.write( "    * σt² = σx² + σy² + 2ρσxσy<BR>\n" );
document.write( "    * σt² = (1.6)² + (1.6)² + 2(0.85)(1.6)(1.6)<BR>\n" );
document.write( "    * σt² = 2.56 + 2.56 + 4.352<BR>\n" );
document.write( "    * σt² = 9.472\r<BR>\n" );
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document.write( "5.  **Standard Deviation of Total Cleaning Time (σt):**<BR>\n" );
document.write( "    * σt = √9.472 ≈ 3.078 hours\r<BR>\n" );
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document.write( "6.  **Distribution of Total Cleaning Time:**<BR>\n" );
document.write( "    * T ~ N(μt = 18.6, σt = 3.078)\r<BR>\n" );
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document.write( "7.  **Calculate the Z-score:**<BR>\n" );
document.write( "    * We want to find P(T < 22.7).<BR>\n" );
document.write( "    * z = (22.7 - 18.6) / 3.078<BR>\n" );
document.write( "    * z = 4.1 / 3.078 ≈ 1.332\r<BR>\n" );
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document.write( "8.  **Find the Probability:**<BR>\n" );
document.write( "    * P(Z < 1.332) using the standard normal distribution table or a calculator.<BR>\n" );
document.write( "    * P(Z < 1.332) ≈ 0.9087\r<BR>\n" );
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document.write( "9.  **Conclusion:**<BR>\n" );
document.write( "    * The probability that the robot will have finished cleaning the ship's hull before the cargo is offloaded and reloaded is approximately 0.9087 or 90.87%.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The probability is approximately 0.9087.<BR>\n" );
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