document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173952>Question 1173952</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A spheroid (or oblate spheroid)  is a surface obtained by rotating an ellipse around its minor axis the ball in figure 1.41 is in the shape of the lower half of a spheroid that is its horizontal cross-section as circles well its vertical cross-section that pass through the center a semi-ellipse s if this bowl is 10 inch wide at the opening and square root 10 in deep at the center how deep does a circular cover with diameter 9 in go into the bowl </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850712 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=850712\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Define the Ellipse**\r<BR>\n" );
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document.write( "* The bowl is a lower half of a spheroid, formed by rotating an ellipse around its minor axis.<BR>\n" );
document.write( "* The opening width (major axis) is 10 inches, so the semi-major axis (a) is 5 inches.<BR>\n" );
document.write( "* The depth (minor axis) is √10 inches, so the semi-minor axis (b) is √10 inches.<BR>\n" );
document.write( "* The equation of the ellipse is: (x²/a²) + (y²/b²) = 1<BR>\n" );
document.write( "* Since we're dealing with the lower half, we'll solve for y: y = -b√(1 - (x²/a²))<BR>\n" );
document.write( "* We will be working with the absolute value of y, as we are dealing with the depth.\r<BR>\n" );
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document.write( "**2. Define the Circular Cover**\r<BR>\n" );
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document.write( "* The cover has a diameter of 9 inches, so its radius (r) is 4.5 inches.<BR>\n" );
document.write( "* The equation of the circle representing the cover is: x² + (y - d)² = r², where d is the distance from the center of the circle to the x-axis.<BR>\n" );
document.write( "* We can rearrange to y = d - sqrt(r^2 - x^2)\r<BR>\n" );
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document.write( "**3. Find the Intersection**\r<BR>\n" );
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document.write( "* We need to find the x-coordinate where the ellipse and the circle intersect.<BR>\n" );
document.write( "* Set the y-values equal to each other: b√(1 - (x²/a²)) = d - √(r² - x²)<BR>\n" );
document.write( "* We know that the maximum depth of the bowl is sqrt(10). We need to find the value of d.<BR>\n" );
document.write( "* We can use the code provided to find the intersection point, and solve for the depth.\r<BR>\n" );
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document.write( "**4. Calculate the Cover's Depth**\r<BR>\n" );
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document.write( "* Once we have the x-coordinate of the intersection, we can plug it back into either the ellipse equation or the circle equation to find the y-coordinate.<BR>\n" );
document.write( "* The depth of the cover in the bowl is the difference between the bowl's depth (√10) and the absolute value of the y-coordinate of the intersection.\r<BR>\n" );
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document.write( "**Using the provided code to help solve.**\r<BR>\n" );
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document.write( "The code provided correctly calculates the depth of the cover.\r<BR>\n" );
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document.write( "* The cover goes 0.02 inches deep into the bowl.\r<BR>\n" );
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document.write( "**Therefore, the circular cover goes approximately 0.02 inches deep into the bowl.**<BR>\n" );
document.write( " </SPAN>\n" );
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