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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 3: Q11E (page 92)

Archaeologists used pieces of burned wood, or charcoal, foundat the site to date prehistoric paintings and drawings on wallsand ceilings of a cave in Lascaux, France. See Figure 3.1.11.Use the information on page 87 to determine the approximateage of a piece of burned wood, if it was found that 85.5% of the C-14 found in living trees of the same type had decayed.

Short Answer

Expert verified

The age of a piece of burned wood is $15,963$ years.

Step by step solution

01

Define carbon dating.

The characteristics of radiocarbon, a radioactive isotope of carbon, are used in radiocarbon dating to determine the age of an object including organic material.

02

Solve for approximate age of a piece of burned wood.

To obtain the age of the piece by the equation in page 87.

$A (t) = A_{0} e^{- \frac{\ln 2}{5 π 20} t}$

As 85.5% of A is decayed and have role="math" localid="1663914672924" $A = 14.5$ , then $A = 0.145 A_{0}$ . The equation becomes,

role="math" localid="1663914718425" $\begin{array}{rcl} 0.145 A_{0} & = & A_{0} e^{- \frac{\ln 2}{37211} t} \\ 0.145 & = & e^{- \frac{\ln 2}{721} t} \\ \ln (0.145) & = & - \frac{\ln 2}{5730} t \\ t & = & \frac{- 5730 \times \ln (0.145)}{\ln (2)} \\ = & 15, 963 years \end{array}$

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Most popular questions from this chapter

Question: When all the curves in a family \(G\left( {x,y,{c_1}} \right) = 0\)intersect orthogonally all the curves in another family\(H\left( {x,y,{c_2}} \right) = 0\), the families are said to be orthogonal trajectories of each other. See Figure 3.R.5. If \(dy/dx = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(dy/dx = - 1/f(x,y)\). In Problems\(\;{\bf{15}} - {\bf{18}}\)find the differential equation of the given family by computing\(dy/dx\)and eliminating \({c_1}\)from this equation. Then find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

\(y = \frac{1}{{x + {c_1}}}\)

Question: Sawing Wood A long uniform piece of wood (cross sections are the same) is cut through perpendicular to its length by a vertical saw blade. See Figure 3.R.6. If the friction between the sides of the saw blade and the wood through which the blade passes is ignored, then it can be assumed that the rate at which the saw blade moves through the piece of wood is inversely proportional to the width of the wood in contact with its cutting edge. As the blade advances through the wood (moving, say, left to right) the width of a cross section changes as a nonnegative continuous function\(w\). If a cross section of the wood is described as a region in the\(xy\)-plane defined over an interval \((a,b)\)then, as shown in Figure 3.R.6(c), the position\(x\)of the saw blade is a function of time \(t\)and the vertical cut made by the blade can be represented by a vertical line segment. The length of this vertical line is the width\(w(x)\)of the wood at that point. Thus the position\(x(t)\)of the saw blade and the rate \(dx/dt\)at which it moves to the right are related to\(w(x)\)by\(w(x)\frac{{dx}}{{dt}} = k,x(0) = a\)

Here\(k\) represents the number of square units of the material removed by the saw blade per unit time.

Suppose the saw is computerized and can be programmed so that\(k = 1\). Find an implicit solution of the foregoing initial-value problem when the piece of wood is a circular\(\log \). Assume a cross section is a circle of radius 2 centered at\((0,0)\) (Hint: To save time see formula 41 in the table of integrals given on the right inside page of the front cover.)
Solve the implicit solution obtained in part (b) for time\(t\)as a function of\(x\). Graph the function\(t(x)\). With the aid of the graph, approximate the time that it takes the saw to cut through this piece of wood. Then find the exact value of this time.

Question: Solve Problem 21 assuming that pure water is pumped into the tank.

(a) Suppose \({\bf{a = b = 1}}\) in the Gompertz differential equation (7). Since the DE is autonomous, use the phase portrait concept of Section \({\bf{2}}{\bf{.1}}\) to sketch representative solution curves corresponding to the cases \({{\bf{P}}_{\bf{0}}}{\bf{ > e}}\) and \({\bf{0 < }}{{\bf{P}}_{\bf{0}}}{\bf{ < e}}\).

(b) Suppose \({\bf{a = 1,b = - 1}}\) in (7). Use a new phase portrait to sketch representative solution curves corresponding to the cases \({{\bf{P}}_{\bf{0}}}{\bf{ > }}{{\bf{e}}^{{\bf{ - 1}}}}\) and \({\bf{0 < }}{{\bf{P}}_{\bf{0}}}{\bf{ < }}{{\bf{e}}^{{\bf{ - 1}}}}\).

(c) Find an explicit solution of (7) subject to \({\bf{P(0) = }}{{\bf{P}}_{\bf{0}}}\).

Suppose that a glass tank has the shape of a cone with circular cross section as shown in Figure 3.2.10. As in part (a) of Problem 33, assume that $h (0) = 2 ft$ corresponds to water filled to the top of the tank, a hole in the bottom is circular with radius $\frac{1}{32}$ in., $g = 32 ft / s^{2}$ , and $c = 0.6$ . Use the differential equation in Problem 12 to find the height h(t) of the water.

(b) Can this water clock measure 12 time intervals of length equal to 1 hour? Explain using sound mathematics.

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