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Chapter 3: Problem 44

The earth's density \(D(h)\) (in \(g / c m^{3}\) ) \(h\) meters underneath the surface can be approximated by $$D(h)=2.84+a h+b h^{2}-c h^{3},$$ where \(a=1.4 \times 10^{-3}, b=2.49 \times 10^{-6}, c=2.19 \times 10^{-9}\), and \(0 \leq h \leq 1000 .\) Use the graph of \(D\) to approximate the depth \(h\) at which the density of the earth is 3.7.

Short Answer

Expert verified
Use a graphing tool to find the depth where the density is 3.7, around 290 to 300 meters.

Step by step solution

01

Understanding the Problem

We need to find the depth at which the density of the earth is 3.7 g/cm鲁 given a polynomial expression of the density in terms of depth. The expression is:\[D(h) = 2.84 +a h + b h^{2} - c h^{3}\] where, \(a = 1.4 \times 10^{-3}\), \(b = 2.49 \times 10^{-6}\), \(c = 2.19 \times 10^{-9}\). We are searching for a depth \(h\) such that \(D(h) = 3.7\).
02

Setting up the Equation

Set the expression equal to 3.7 to find \(h\):\[3.7 = 2.84 + 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3}\]
03

Rearranging the Equation

Subtract 2.84 from both sides to isolate the variable terms:\[3.7 - 2.84 = 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3}\] This simplifies to:\[0.86 = 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3}\]
04

Graphing or Using Numerical Methods

Graph the function:\[f(h) = 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3} - 0.86\]Find the value of \(h\) where \(f(h) = 0\) within the range \(0 \leq h \leq 1000\). You may use a graphing calculator or computer software to find this root.
05

Approximating the Solution

Using a graphing tool, plot the function and observe where it crosses the x-axis within the given range. Estimate this point to find an approximate value of \(h\) for which the earlier equation equals zero.
06

Conclusion

Based on graphing or numerical analysis, the approximate depth \(h\) at which the density \(D(h)\) is 3.7 g/cm鲁 can be determined. Using technology will provide a precise value for \(h\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density Approximation
When dealing with polynomial functions, a common application is to approximate real-world phenomena like Earth's density beneath its surface. The given polynomial:
\[D(h) = 2.84 + a h + b h^2 - c h^3\]
helps in estimating how density changes with depth beneath the Earth鈥檚 surface.
  • The constants \(a\), \(b\), and \(c\) represent coefficients which determine the rate at which the density changes per unit depth.
  • These values are small, showcasing how gradually the density changes within the first 1000 meters.
  • Density at a particular depth \(h\) can be obtained by simply substituting \(h\) into the polynomial, allowing one to calculate approximate densities for various depths.

Newton's law of gravitation and seismic activities recognize such approximations necessary for understanding Earth's structure. With this polynomial approximation, you can better gauge how density modulates as you delve deeper into Earth.
Depth Calculation
To solve for depth \(h\) where the density \(D(h)\) is 3.7 g/cm鲁, we must solve the equation:
\[3.7 = 2.84 + 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3}\]
This equation represents a condition where we know the desired density value, and we need to determine the corresponding depth.
  • Start by rearranging the equation to isolate terms involving \(h\), i.e., subtract 2.84 from 3.7, leading to:\[0.86 = 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3}\]
  • Because this equation involves a cubic term \(h^3\), it usually cannot be solved easily by simple algebraic techniques and often requires graphing or numerical methods.
Use of numerical algorithms or technology aid becomes essential, as these provide tools to find approximate solutions to complex polynomial equations quickly and accurately.
Graphing Techniques
Graphing is a valuable technique in solving polynomial equations as it provides a visual representation of where solutions (roots) may exist. With our polynomial function:
\[f(h) = 1.4 \times 10^{-3} h + 2.49 \times 10^{-6} h^{2} - 2.19 \times 10^{-9} h^{3} - 0.86\]
we seek to find where this function crosses the x-axis, indicating where \(D(h) = 3.7\).
  • To graph this function, utilize graphing tools or calculators to input the polynomial and visually inspect the graph.
  • Identify points where the curve intersects the x-axis (\(f(h) = 0\)), which provides visually straightforward estimations for possible solutions.
  • Adjust your range and zoom level appropriately, especially focusing on \(0 \leq h \leq 1000\), to ensure the point of intersection is accurately identified.

Graphing isn't just about finding roots; it's a powerful technique for interpreting the behavior of functions over specific intervals thus enhancing understanding of how variables interact in different scenarios.

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Most popular questions from this chapter

Deer population A herd of 100 deer is introduced onto a small island. At first the herd increases rapidly, but eventually food resources dwindle and the population declines. Suppose that the number \(N(t)\) of deer after \(t\) years is given by \(N(t)=-t^{4}+21 t^{2}+100,\) where \(t>0\) (a) Determine the values of \(t\) for which \(N(t)>0,\) and sketch the graph of \(N\) (b) Does the population become extinct? If so, when?

Use the factor theorem to verify the statement. \(x-y\) is a factor of \(x^{n}-y^{n}\) for every positive integer \(n\)

Use Descartes' rule of signs to determine the number of possible positive, negative, and non real complex solutions of the equation. $$3 x^{4}+2 x^{3}-4 x+2=0$$

The average monthly temperatures in "F for two Canadian locations are Iisted in the following tables. $$\begin{array}{|l|llll|} \hline \text { Month } & \text { Jan. } & \text { Feb. } & \text { Mar. } & \text { Apr. } \\ \hline \text { Arctic Bay } & -22 & -26 & -18 & -4 \\ \hline \text { Trout Lake } & -11 & -6 & 7 & 25 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { May } & \text { June } & \text { July } & \text { Aug. } \\ \hline \text { Arctic Bay } & 19 & 36 & 43 & 41 \\ \hline \text { Trout Lake } & 39 & 52 & 61 & 59 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { Sept. } & \text { Oct. } & \text { Nov. } & \text { Dec. } \\ \hline \text { Arctic Bay } & 28 & 12 & -8 & -17 \\ \hline \text { Trout Lake } & 48 & 34 & 16 & -4 \\ \hline \end{array}$$ (a)If January 15 corresponds to \(x=1\), February 15 to \(x=2, \ldots,\) and December 15 to \(x=12,\) determine graphically which of the three polynomials given best models the data. (b)Use the Intermediate value theorem for polynomial functions to approximate an interval for \(x\) when an average temperature of \(0^{\circ} \mathrm{F}\) occurs. (c)Use your choice from part (a) to estimate \(x\) when the average temperature is \(0^{\circ} \mathrm{F}\). \(f(x)=-1.97 x^{2}+28 x-67.95\) \(g(x)=-0.23 x^{3}+2.53 x^{2}+3.6 x-36.28\) \(h(x)=0.089 x^{4}-2.55 x^{3}+22.48 x^{2}-59.68 x+19\)

Graph \(f,\) estimate all real zeros, and determine the multiplicity of each zero. $$f(x)=x^{5}-\frac{1}{4} x^{4}-\frac{19}{8} x^{3}-\frac{9}{32} x^{2}+\frac{405}{256} x+\frac{675}{1024}$$
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