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Chapter 5: Problem 3

Assume that a long horizontal pipe connects the bottom of a reservoir with a drainage area. Cox's formula provides a way of determining the velocity \(v\) of the water flowing through the pipe: $$ \frac{H d}{L}=\frac{4 v^{2}+5 v-2}{1200} . $$ Here \(H\) is the depth of the reservoir in feet, \(d\) is the pipe diameter in inches, \(L\) is the length of the pipe in feet, and the velocity \(v\) of the water is in feet per second. (See Figure \(5.103\) on the following page.) a. Graph the quadratic function \(4 v^{2}+5 v-2\) using a horizontal span from 0 to 10 . b. Judging on the basis of Cox's formula, is it possible to have a velocity of \(0.25\) foot per second? c. Find the velocity of the water in the pipe if its diameter is 4 inches, its length is 1000 feet, and the reservoir is 50 feet deep. d. If the water velocity is too high, there will be erosion problems. Assuming that the pipe length is 1000 feet and the reservoir is 50 feet deep, determine the largest pipe diameter that will ensure that the water velocity does not exceed 10 feet per second.

Short Answer

Expert verified
a. Graph \(y = 4v^2 + 5v - 2\). b. Confirm if \(\frac{Hd}{L} > 0\) for \(v = 0.25\). c. Solve for \(v\) using given values. d. Find \(d\) ensuring \(v \leq 10\) ft/s.

Step by step solution

01

Graph the Function

To graph the function \(f(v) = 4v^2 + 5v - 2\), plot it over the interval \(0 \leq v \leq 10\). This function is a parabola opening upwards as the coefficient of \(v^2\) is positive. For detailed graphing, calculate some values by substituting \(v\) values between 0 and 10 and plot the points. The vertex form or the quadratic formula might help identify key points like its vertex and intercepts.
02

Determine Feasibility of Velocity 0.25

For velocity \(v = 0.25 \text{ ft/s}\), substitute \(v\) into the function \(4v^2 + 5v - 2\) to calculate \(f(0.25)\). Then, use Cox's formula and solve \(\frac{Hd}{L} = \frac{f(0.25)}{1200}\). If \(f(0.25)\) provides a positive solution for \(\frac{Hd}{L}\), then a velocity of 0.25 ft/s is possible.
03

Solve for Water Velocity

Substitute \(d = 4 \text{ inches}\), \(L = 1000 \text{ feet}\), and \(H = 50 \text{ feet}\) into Cox鈥檚 formula: \(\frac{Hd}{L} = \frac{4v^2 + 5v - 2}{1200}\). Therefore, \(\frac{50 \times 4}{1000} = \frac{4v^2 + 5v - 2}{1200}\). Simplify the equation and solve the resulting quadratic equation to find \(v\).
04

Determine Maximum Diameter for Safe Velocity

Assuming \(v = 10 \text{ ft/s}\), substitute \(L = 1000 \text{ feet}\) and \(H = 50 \text{ feet}\) into Cox鈥檚 formula. Rearrange \(\frac{Hd}{L} = \frac{4 \times 10^2 + 5 \times 10 - 2}{1200}\) to solve for \(d\). Solve \(\frac{50d}{1000} = \frac{452}{1200}\) to find the largest diameter \(d\) that keeps the velocity at or below 10 ft/s.

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Key Concepts

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

Cox's Formula
Cox's formula is crucial in calculating the water flow velocity in a pipe that is connected to a reservoir. The formula you've encountered here, \( \frac{H d}{L}=\frac{4 v^{2}+5 v-2}{1200} \), provides the relationship between different parameters involved in the water flow. This equation effectively combines:
  • Reservoir Depth \( H \)
  • Pipe Diameter \( d \)
  • Pipe Length \( L \)
  • Water Flow Velocity \( v \)

Understanding Cox's formula involves recognizing how these parameters are interrelated. For instance, increasing the reservoir depth \( H \) or pipe diameter \( d \) increases the left side of the equation, which implies a potential increase in water velocity. Conversely, increasing pipe length \( L \) reduces the left-hand side, suggesting decreased water velocity. This formula is particularly useful because it simplifies calculating parts of water systems using basic measurable quantities.
Water Flow Velocity
Calculating water flow velocity \( v \) accurately is vital, as it determines the efficiency and safety of water delivery systems. Velocity in the context of Cox's formula affects both stability and erosion potential. The quadratic function \( 4v^2 + 5v - 2 \) describes the relationship between velocity and the other terms in the formula.

A velocity that is too high may cause erosion damage, while too low a velocity could indicate inefficiencies. When using this formula, solve for \( v \) by substituting measured values鈥攖he depth of the reservoir, pipe diameter, and length of the pipe鈥攊nto the equation. This requires solving a quadratic equation, where finding the roots provides potential water velocities under given conditions.

Graphing \( 4v^2 + 5v - 2 \) helps visualize how changes in velocity are expressed in the formula. It's a parabola opening upwards, meaning it has a minimum value, which is key in understanding how velocity behaves within the given parameters.
Reservoir Depth
Reservoir depth \( H \) is an integral part of determining water flow through pipes. It directly influences the potential energy available to drive the water. In Cox's formula, increasing \( H \) tends to increase the water flow velocity \( v \), as more water weight pushes through the pipe.

The reservoirs act as a water source and a buffer. Deeper reservoirs typically mean a higher pressure at the exit point of the pipe, resulting in higher velocities. This can be beneficial in ensuring effective water delivery, but excessive velocity might risk structural damage or accelerated material wear. Therefore, optimizing \( H \) is important both for efficiency and sustainability of the water system.

In practical applications, controlling reservoir depth also lets engineers regulate outflows and manage resources efficiently.
Pipe Diameter
The diameter of the pipe \( d \) used in Cox's formula is a critical factor affecting water flow. A larger diameter allows for more water to flow through the pipe, which can effectively reduce resistance and thus increase the velocity, provided all other factors remain constant.

One of the exercises posed is determining the largest safe diameter to prevent exceeding a target velocity like 10 feet per second. Thus, as a key design element, diameter must balance between being wide enough to carry the necessary water volume without incurring issues such as erosion.

The role of diameter
  • Increases in diameter can imply higher capacity but could result in higher costs.
  • Too large a diameter, with increased velocities, may not be sustainable in terms of material integrity over time.
  • The choice of diameter must also relate to the context鈥攃onsidering space, existing infrastructure, and cost.
Well-considered digital or physical models can aid in visualizing and testing various diameters before implementation.

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Most popular questions from this chapter

Roughly \(90 \%\) of all stars are main-sequence stars. Exceptions include supergiants, giants, and dwarfs. For main-sequence stars (including the sun) there is an important relationship called the mass-luminosity relation between the relative luminosity \({ }^{25} L\) and the mass \(M\) in terms of solar masses. Relative masses and luminosities of several main-sequence stars are reported in the accompanying table. a. Find a power model for the data in this table. (Round the power and the coefficient to one decimal place.) The function you find is known to astronomers as the mass-luminosity relation. b. Kruger 60 is a main-sequence star that is about \(0.11\) solar mass. Use functional notation to express the relative luminosity of Kruger 60 , and then calculate that value. $$ \begin{array}{|l|c|c|} \hline \text { Star } & \begin{array}{c} \text { Solar mass } \\ M \end{array} & \begin{array}{c} \text { Luminosity } \\ L \end{array} \\ \hline \text { Spica } & 7.3 & 1050 \\ \hline \text { Vega } & 3.1 & 55 \\ \hline \text { Altair } & 1 & 1.1 \\ \hline \text { The Sun } & 1 & 1 \\ \hline 61 \text { Cygni A } & 0.17 & 0.002 \\ \hline \end{array} $$ c. Wolf 359 has a relative luminosity of about \(0.0001\). How massive is Wolf 359 ? d. If one star is 3 times as massive as another, how do their luminosities compare?

Table \(5.3\) gives the length \(L\), in inches, of a flying animal and its maximum speed \(F\), in feet per second, when it flies. \({ }^{27}\) (For comparison, 10 feet per second is about \(6.8\) miles per hour.) $$ \begin{array}{|l|c|c|} \hline \text { Animal } & \text { Length } L & \begin{array}{c} \text { Flying speed } \\ F \end{array} \\ \hline \text { Fruit fly } & 0.08 & 6.2 \\ \hline \text { Horse fly } & 0.51 & 21.7 \\ \hline \begin{array}{l} \text { Ruby-throated } \\ \text { hummingbird } \end{array} & 3.2 & 36.7 \\ \hline \text { Willow warbler } & 4.3 & 39.4 \\ \hline \text { Flying fish } & 13 & 51.2 \\ \hline \text { Bewick's swan } & 47 & 61.7 \\ \hline \text { White pelican } & 62 & 74.8 \\ \hline \end{array} $$ a. Judging on the basis of this table, is it generally true that larger animals fly faster? b. Find a formula that models \(F\) as a power function of \(L\). c. Make the graph of the function in part b. d. Is the graph you found in part c concave up or concave down? Explain in practical terms what your answer means. e. If one bird is 10 times longer than another, how much faster would you expect it to fly? (Use the homogeneity property of power functions.)

It is a consequence of Newton's law of gravitation that near the surface of any planet, the distance \(D\) fallen by a rock in time \(t\) is given by \(D=c t^{2}\). That is, distance fallen is proportional to the square of the time, no matter what planet one may be on. But the value of \(c\) depends on the mass of the planet. For Earth, if time is measured in seconds and distance in feet, the value of \(c\) is 16 . a. Suppose a rock is falling near the surface of a planet. What is the comparison in distance fallen from 2 seconds to 6 seconds into the drop? (Hint: This question may be rephrased as follows: "If time increases by a factor of 3 , by what factor will distance increase?") b. For objects falling near the surface of Mars, if time is measured in seconds and distance in feet, the value of \(c\) is \(6.4\). If a rock is dropped from 70 feet above the surface of Mars, how long will it take for the rock to strike the ground? c. On Venus, a rock dropped from 70 feet above the surface takes \(2.2\) seconds to strike the ground. What is the value of \(c\) for Venus?

In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypes-that is, instances in which the two alleles carried at a particular site on an individual's chromosomes are both the same. For populations in which bloodrelated individuals mate, there is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. In general, the frequency of homozygous children from matings of blood-related parents is greater than that for children from unrelated parents. \(^{67}\) Measured over a large number of generations, the proportion of heterozygous genotypes-that is, nonhomozygous genotypes-changes by a constant factor \(\lambda_{1}\) from generation to generation. The factor \(\lambda_{1}\) is a number between 0 and 1 . If \(\lambda_{1}=0.75\), for example, then the proportion of heterozygous individuals in the population decreases by \(25 \%\) in each generation. In this case, after 10 generations the proportion of heterozygous individuals in the population decreases by \(94.37 \%\), since \(0.75^{10}=\) \(0.0563\), or \(5.63 \%\). In other words, \(94.37 \%\) of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For matings between siblings, \(\lambda_{1}\) can be determined as the largest value of \(\lambda\) for which $$ \lambda^{2}=\frac{1}{2} \lambda+\frac{1}{4}. $$ This equation comes from carefully accounting for the genotypes for the present generation (the \(\lambda^{2}\) term) in terms of those of the previous two generations (represented by \(\lambda\) for the parents' generation and by the constant term for the grandparents' generation). a. Find both solutions to the quadratic equation above and identify which is \(\lambda_{1}\). (Use a horizontal span from \(-1\) to 1 in this exercise and the following two exercises.) b. After 5 generations what proportion of the population will be homozygous? c. After 20 generations what proportion of the population will be homozygous?

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