document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163696>Question 1163696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It is believed that two quantities, z and d are Connected by the relationship of the form z=kd^n where k and n are provided  that d doesn't exceed some fixed (but unknown)values D.An experiment produced the following data<BR>\n" );
document.write( "D   750    810   870   930   990   1050   1110    1170<BR>\n" );
document.write( "Z   2.1   2.6    3.2   4.0   4.8    5.6    5.9     6.1<BR>\n" );
document.write( "a)Plot the values of log10Z against log10 d.Use these points to suggest a <BR>\n" );
document.write( "   value for D.<BR>\n" );
document.write( "b)It is known tht for d < D,n is a whole number.Use your graph to find the value of n.Show also that k=5×10^-9<BR>\n" );
document.write( "c) Use your value of n and the estimate k=5×10^-9 to find the value of d for which z=30. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854372 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=854372\">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 numpy as np<BR>\n" );
document.write( "import matplotlib.pyplot as plt\r<BR>\n" );
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document.write( "d = np.array([750, 810, 870, 930, 990, 1050, 1110, 1170])<BR>\n" );
document.write( "z = np.array([2.1, 2.6, 3.2, 4.0, 4.8, 5.6, 5.9, 6.1])\r<BR>\n" );
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document.write( "log_d = np.log10(d)<BR>\n" );
document.write( "log_z = np.log10(z)\r<BR>\n" );
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document.write( "plt.scatter(log_d, log_z, color='blue', label='Data points')<BR>\n" );
document.write( "plt.plot(log_d, log_z, 'b--', alpha=0.3)<BR>\n" );
document.write( "plt.xlabel('log10(d)')<BR>\n" );
document.write( "plt.ylabel('log10(z)')<BR>\n" );
document.write( "plt.title('log10(z) vs log10(d)')<BR>\n" );
document.write( "plt.grid(True)<BR>\n" );
document.write( "plt.savefig('log_plot.png')\r<BR>\n" );
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document.write( "# Calculate slopes between consecutive points<BR>\n" );
document.write( "slopes = np.diff(log_z) / np.diff(log_d)<BR>\n" );
document.write( "print(f\"Slopes: {slopes}\")\r<BR>\n" );
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document.write( "# Let's perform linear regression for the first few points to check n<BR>\n" );
document.write( "# Trying first 5 points (d = 750 to 990)<BR>\n" );
document.write( "for i in range(3, 8):<BR>\n" );
document.write( "    subset_log_d = log_d[:i]<BR>\n" );
document.write( "    subset_log_z = log_z[:i]<BR>\n" );
document.write( "    m, c = np.polyfit(subset_log_d, subset_log_z, 1)<BR>\n" );
document.write( "    print(f\"Points 1 to {i}: slope (n) = {m:.4f}, intercept (log10 k) = {c:.4f}, k = {10**c:.2e}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Slopes: [2.7750937  2.90571474 3.34591322 2.91619507 2.61980574 0.93910191<BR>\n" );
document.write( " 0.63324575]<BR>\n" );
document.write( "Points 1 to 3: slope (n) = 2.8372, intercept (log10 k) = -7.8355, k = 1.46e-08<BR>\n" );
document.write( "Points 1 to 4: slope (n) = 2.9825, intercept (log10 k) = -8.2565, k = 5.54e-09<BR>\n" );
document.write( "Points 1 to 5: slope (n) = 3.0028, intercept (log10 k) = -8.3155, k = 4.84e-09<BR>\n" );
document.write( "Points 1 to 6: slope (n) = 2.9634, intercept (log10 k) = -8.2005, k = 6.30e-09<BR>\n" );
document.write( "Points 1 to 7: slope (n) = 2.7732, intercept (log10 k) = -7.6442, k = 2.27e-08\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "[image-tag: code-generated-image-0-1776447487869153342]\r<BR>\n" );
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document.write( "To find the values of $k$, $n$, and $D$ for the relationship $z = k d^n$, we transform the equation by taking the logarithm (base 10) of both sides:<BR>\n" );
document.write( "$$\log_{10} z = n \log_{10} d + \log_{10} k$$<BR>\n" );
document.write( "This is a linear equation of the form $y = mx + c$, where the gradient is $n$ and the y-intercept is $\log_{10} k$.\r<BR>\n" );
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document.write( "### a) Plotting and suggesting a value for $D$\r<BR>\n" );
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document.write( "First, we calculate the values for $\log_{10} d$ and $\log_{10} z$:\r<BR>\n" );
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document.write( "| $d$ | $750$ | $810$ | $870$ | $930$ | $990$ | $1050$ | $1110$ | $1170$ |<BR>\n" );
document.write( "| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |<BR>\n" );
document.write( "| $z$ | $2.1$ | $2.6$ | $3.2$ | $4.0$ | $4.8$ | $5.6$ | $5.9$ | $6.1$ |<BR>\n" );
document.write( "| $\log_{10} d$ | $2.875$ | $2.908$ | $2.939$ | $2.968$ | $2.996$ | $3.021$ | $3.045$ | $3.068$ |<BR>\n" );
document.write( "| $\log_{10} z$ | $0.322$ | $0.415$ | $0.505$ | $0.602$ | $0.681$ | $0.748$ | $0.771$ | $0.785$ |\r<BR>\n" );
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document.write( "**Analysis of the Plot:**<BR>\n" );
document.write( "When plotting these points, we look for where the linear trend stops. Calculating the gradient (slope) between successive points shows:<BR>\n" );
document.write( "* From $d=750$ to $d=1050$, the slope remains consistently near **$3.0$**.<BR>\n" );
document.write( "* After $d=1050$, the slope drops drastically to $\approx 0.96$ (between 1050-1110) and $\approx 0.63$ (between 1110-1170).\r<BR>\n" );
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document.write( "**Suggested Value for $D$:**<BR>\n" );
document.write( "Based on the graph, the linear relationship holds up to **$D \approx 1050$**. Beyond this value, the data points clearly deviate from the straight line.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### b) Finding $n$ and showing $k = 5 \times 10^{-9}$\r<BR>\n" );
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document.write( "**Determining $n$:**<BR>\n" );
document.write( "Since $n$ is known to be a whole number for $d < D$, we look at the slopes of the linear portion:<BR>\n" );
document.write( "* Slope ($870$ to $930$) $\approx \frac{0.602 - 0.505}{2.968 - 2.939} = \frac{0.097}{0.029} \approx 3.3$<BR>\n" );
document.write( "* Slope ($930$ to $990$) $\approx \frac{0.681 - 0.602}{2.996 - 2.968} = \frac{0.079}{0.028} \approx 2.8$<BR>\n" );
document.write( "The average slope is approximately 3. Thus, **$n = 3$**.\r<BR>\n" );
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document.write( "**Showing $k = 5 \times 10^{-9}$:**<BR>\n" );
document.write( "Using the formula $k = \frac{z}{d^3}$ for values where $d < D$:<BR>\n" );
document.write( "* For $d=750$: $k = \frac{2.1}{750^3} \approx 4.98 \times 10^{-9}$<BR>\n" );
document.write( "* For $d=930$: $k = \frac{4.0}{930^3} \approx 4.97 \times 10^{-9}$<BR>\n" );
document.write( "* For $d=990$: $k = \frac{4.8}{990^3} \approx 4.95 \times 10^{-9}$\r<BR>\n" );
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document.write( "Rounding to one significant figure, we find **$k = 5 \times 10^{-9}$**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c) Find the value of $d$ for which $z=30$\r<BR>\n" );
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document.write( "Using the model $z = (5 \times 10^{-9})d^3$, we substitute $z = 30$:<BR>\n" );
document.write( "$$30 = (5 \times 10^{-9}) d^3$$<BR>\n" );
document.write( "$$d^3 = \frac{30}{5 \times 10^{-9}}$$<BR>\n" );
document.write( "$$d^3 = 6 \times 10^9$$<BR>\n" );
document.write( "$$d = \sqrt[3]{6 \times 10^9}$$<BR>\n" );
document.write( "$$d = 1000 \times \sqrt[3]{6}$$<BR>\n" );
document.write( "$$d \approx 1000 \times 1.817$$<BR>\n" );
document.write( "**$d \approx 1817$**\r<BR>\n" );
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document.write( "*(Note: While $1817 > D$, the calculation based strictly on the suggested constants yields this result.)* </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );