document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200771>Question 1200771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The dimensions of the square grids in the grid pattern are 1 × 1. What is the minimum number of such grids that need to be selected so that when they are randomly stacked on top of each other, a water droplet of radius r falling from the top would be absorbed by the surface with a probability of at least α? We assume that the droplet is absorbed if it hits at least one grid. If we stack this number of grids and a stream falls on it consisting of a% of r radius droplets and (100 - a)% of s radius droplets, what fraction of the grids would remain uncovered? ([α, r, s, a] = [0.72, 0.1, 0.13, 21]) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848158 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the Minimum Number of Grids for a Single Droplet**\r<BR>\n" );
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document.write( "* **Probability of Absorption by a Single Grid:**<BR>\n" );
document.write( "    * The droplet will be absorbed by a single grid if its center falls within a circle of radius r + 0.5 (since the grid size is 1x1 and the droplet has radius r).<BR>\n" );
document.write( "    * Area of the circle: π * (r + 0.5)² <BR>\n" );
document.write( "    * Area of the grid: 1 * 1 = 1<BR>\n" );
document.write( "    * Probability of absorption by a single grid: (π * (r + 0.5)²) / 1 = π * (r + 0.5)²\r<BR>\n" );
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document.write( "* **Probability of Absorption by 'n' Grids (Cumulative Probability):**<BR>\n" );
document.write( "    * Assuming independent events, the probability of the droplet not being absorbed by any of the 'n' grids is:<BR>\n" );
document.write( "        * (1 - Probability of absorption by a single grid)^n <BR>\n" );
document.write( "        * = (1 - π * (r + 0.5)²) ^ n\r<BR>\n" );
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document.write( "    * Therefore, the probability of the droplet being absorbed by at least one of the 'n' grids is:<BR>\n" );
document.write( "        * 1 - (1 - π * (r + 0.5)²) ^ n\r<BR>\n" );
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document.write( "* **Find 'n' for Desired Probability (α):**<BR>\n" );
document.write( "    * 1 - (1 - π * (r + 0.5)²) ^ n ≥ α<BR>\n" );
document.write( "    * (1 - π * (r + 0.5)²) ^ n ≤ 1 - α<BR>\n" );
document.write( "    * n * log(1 - π * (r + 0.5)²) ≤ log(1 - α) <BR>\n" );
document.write( "    * n ≥ log(1 - α) / log(1 - π * (r + 0.5)²)\r<BR>\n" );
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document.write( "* **Substitute given values:**<BR>\n" );
document.write( "    * α = 0.72, r = 0.1<BR>\n" );
document.write( "    * n ≥ log(1 - 0.72) / log(1 - π * (0.1 + 0.5)²) <BR>\n" );
document.write( "    * n ≥ log(0.28) / log(1 - π * 0.36) <BR>\n" );
document.write( "    * n ≥ 2.04 \r<BR>\n" );
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document.write( "* **Since 'n' must be an integer, the minimum number of grids is 3.**\r<BR>\n" );
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document.write( "**2. Calculate Fraction of Uncovered Grids for the Stream**\r<BR>\n" );
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document.write( "* **Probability of a droplet of radius 'r' being absorbed:** 0.72 (given)<BR>\n" );
document.write( "* **Probability of a droplet of radius 's' being absorbed:**<BR>\n" );
document.write( "    * Calculate using the same formula as in step 1:<BR>\n" );
document.write( "        * 1 - (1 - π * (s + 0.5)²) ^ 3 <BR>\n" );
document.write( "        * 1 - (1 - π * (0.13 + 0.5)²) ^ 3 <BR>\n" );
document.write( "        * ≈ 0.836\r<BR>\n" );
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document.write( "* **Probability of a droplet NOT being absorbed:**<BR>\n" );
document.write( "    * Droplet of radius 'r': 1 - 0.72 = 0.28<BR>\n" );
document.write( "    * Droplet of radius 's': 1 - 0.836 = 0.164\r<BR>\n" );
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document.write( "* **Fraction of uncovered grids:**<BR>\n" );
document.write( "    * (0.21 * 0.28) + (0.79 * 0.164) ≈ 0.217\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **Minimum number of grids:** 3<BR>\n" );
document.write( "* **Fraction of uncovered grids:** 0.217\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This calculation assumes that the droplets fall independently and uniformly on the grid surface.<BR>\n" );
document.write( "* This analysis provides an estimate. Actual results may vary depending on the specific distribution of droplets within the stream.<BR>\n" );
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