document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163329>Question 1163329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If you are in shallow water and look at an object below the surface of the water, <BR>\n" );
document.write( "the object will look farther away from you than it really is. This is because as light passes <BR>\n" );
document.write( "through air and water it bends or refracts the light rays. The index of refraction for water is 1.33.<BR>\n" );
document.write( "This is the ratio of the sine of theta 1 and the sine of theta 2.\r<BR>\n" );
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document.write( "a)You are in 2 feet of water in the shallow end of a pool. You look down at some goggles at angle <BR>\n" );
document.write( "(theta=60) degrees measured from a line perpendicular to the surface of the water. Find (theta 2 ) \r<BR>\n" );
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document.write( "b) Find the distances x and y <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 #854382 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=854382\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import math\r<BR>\n" );
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document.write( "# Given values<BR>\n" );
document.write( "n_air = 1.0<BR>\n" );
document.write( "n_water = 1.33<BR>\n" );
document.write( "theta_1_deg = 60.0<BR>\n" );
document.write( "depth_y = 2.0 # feet\r<BR>\n" );
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document.write( "# Part a: Find theta_2<BR>\n" );
document.write( "# n1 * sin(theta1) = n2 * sin(theta2)<BR>\n" );
document.write( "theta_1_rad = math.radians(theta_1_deg)<BR>\n" );
document.write( "sin_theta_2 = (n_air * math.sin(theta_1_rad)) / n_water<BR>\n" );
document.write( "theta_2_rad = math.asin(sin_theta_2)<BR>\n" );
document.write( "theta_2_deg = math.degrees(theta_2_rad)\r<BR>\n" );
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document.write( "# Part b: Find distances x and y<BR>\n" );
document.write( "# y is the actual depth<BR>\n" );
document.write( "y = depth_y<BR>\n" );
document.write( "# x is the horizontal distance from the entry point to the object<BR>\n" );
document.write( "# From the right triangle in the water: tan(theta_2) = x / y<BR>\n" );
document.write( "x = y * math.tan(theta_2_rad)\r<BR>\n" );
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document.write( "print(f\"{theta_1_deg=}\")<BR>\n" );
document.write( "print(f\"{theta_2_deg=}\")<BR>\n" );
document.write( "print(f\"{y=}\")<BR>\n" );
document.write( "print(f\"{x=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "theta_1_deg=60.0<BR>\n" );
document.write( "theta_2_deg=40.62813065148206<BR>\n" );
document.write( "y=2.0<BR>\n" );
document.write( "x=1.715911347328087\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To solve for the angle of refraction and the geometric distances, we use **Snell's Law**, which describes how light bends when crossing the boundary between two different media (in this case, air and water).\r<BR>\n" );
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document.write( "### (a) Find $\theta_2$<BR>\n" );
document.write( "Snell's Law is given by the formula:<BR>\n" );
document.write( "$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$<BR>\n" );
document.write( "Where:<BR>\n" );
document.write( "* $n_1$ is the index of refraction for air ($\approx 1.00$).<BR>\n" );
document.write( "* $n_2$ is the index of refraction for water ($1.33$).<BR>\n" );
document.write( "* $\theta_1$ is the angle of incidence in air ($60^\circ$).<BR>\n" );
document.write( "* $\theta_2$ is the angle of refraction in water.\r<BR>\n" );
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document.write( "Using the ratio provided in the problem ($1.33 = \frac{\sin \theta_1}{\sin \theta_2}$):<BR>\n" );
document.write( "$$1.33 = \frac{\sin 60^\circ}{\sin \theta_2}$$<BR>\n" );
document.write( "$$\sin \theta_2 = \frac{\sin 60^\circ}{1.33}$$<BR>\n" );
document.write( "$$\sin \theta_2 = \frac{0.8660}{1.33} \approx 0.6511$$<BR>\n" );
document.write( "$$\theta_2 = \arcsin(0.6511) \approx 40.63^\circ$$\r<BR>\n" );
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document.write( "**The angle of refraction $\theta_2$ is approximately $40.63^\circ$.**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (b) Find the distances $x$ and $y$<BR>\n" );
document.write( "In this geometric model of refraction:<BR>\n" );
document.write( "* **$y$** represents the vertical depth of the water. According to the problem, you are in 2 feet of water.<BR>\n" );
document.write( "    $$y = 2\text{ ft}$$<BR>\n" );
document.write( "* **$x$** represents the horizontal distance from the point where the light ray enters the water to the location of the goggles on the pool floor.\r<BR>\n" );
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document.write( "Using the right triangle formed by the light ray under the water:<BR>\n" );
document.write( "$$\tan \theta_2 = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}$$<BR>\n" );
document.write( "$$x = y \cdot \tan \theta_2$$<BR>\n" );
document.write( "$$x = 2 \cdot \tan(40.63^\circ)$$<BR>\n" );
document.write( "$$x \approx 2 \cdot 0.8580$$<BR>\n" );
document.write( "$$x \approx 1.716\text{ ft}$$\r<BR>\n" );
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document.write( "**Final Results:**<BR>\n" );
document.write( "* **$\theta_2 \approx 40.63^\circ$**<BR>\n" );
document.write( "* **$y = 2\text{ ft}$** (Actual depth)<BR>\n" );
document.write( "* **$x \approx 1.72\text{ ft}$** (Horizontal distance) </SPAN>\n" );
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