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Chapter 3: Q11E (page 102)

Leaking Cylindrical Tank A tank in the form of a right circular cylinder standing on its end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by

whereandare the cross-sectional areas of the water and the hole, respectively.

(a) Solve the DE if the initial height of the water isBy hand, sketch the graph ofand give its interval I of definition in terms of the symbolsand H. Use

(b) Suppose the tank is 10 feet high and has a radius of 2 feet and the circular hole has a radiusinch. If the tank is initially full, how long will it take to empty?

Short Answer

Expert verified

Answer

The graph of and give its interval of definition in terms of the symbols, and is formed.

The time taken for the tank to be empty is approximately

Step by step solution

01

Define the equation of height of the water

The height of water in this tank is described by the

Where is the cross-sectional area of water, with the condition.

02

Find the amount of water in the tank at time t.

(a).

When

Then we have found

03

Find the constant.

Put the point of condition

Then we have

Now put the value of C3 in equation 1

………….. (2)

This is the height of the water inside the cylinder at time t.

Now sketch the graph by using the point

04

Find the time at which the tank is empty when 

(b)

When

Then,

The time taken for the tank to be empty is approximately.

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Most popular questions from this chapter

Inverted Conical Tank Suppose that the conical tank in Problem 13(a) is inverted, as shown in Figure 3.2.5, and that water leaks out a circular hole of radius 2 inches in the center of its circular base. Is the time it takes to empty a full tank the same as for the tank with vertex down in Problem 13? Take the friction/contraction coefficient to be c=0.6and g=32ft/s2

Potassium-40 Decay The chemical element potassium is a soft metal that can be found extensively throughout the Earth's crust and oceans. Although potassium occurs naturally in the form of three isotopes, only the isotope potassium-40 (K-40)is radioactive. This isotope is also unusual in that it decays by two different nuclear reactions. Over time, by emitting beta particles a great percentage of an initial amount K0of K-40 decays into the stable isotope calcium-40 (Ca40), whereas by electron capture a smaller percentage of decays into the stable isotope argon-40 (Ar-40). Because the rates at which the amounts C(t)of Ca-40 and A(t)of Ar-40increase are proportional to the amount K(t)of potassium present, and the rate at which K(t)decays is also proportional to K(t), we obtain the system of linear first-order equations

dCdt=λ1K

dAdt=λ2K

dKdt=-λ1+λ2K

where λ1and λ2are positive constants of proportionality. By proceeding as in Problem 1 we can solve the foregoing mathematical model.

(a) From the last equation in the given system of differential equations find K(t)if K(0)=K0. Then useto findandfrom the first and second equations. Assume that C(0)=0and A(0)=0.

(b) It is known that λ1=4.7526×10-10and λ2=0.5874×10-10. Find the half-life of K-40.

(c) Use C(t)and A(t)found in part (a) to determine the percentage of an initial amountK0of K-40that decays into Ca-40and the percentage that decays into Ar-40over a very long period of time.

When a vertical beam of light passes through a transparent medium, the rate at which its intensity Idecreases is proportional toI(t), where trepresents the thickness of the medium (in feet).In clear seawater, the intensity 3feet below the surface is 25% of the initial intensityI0of the incident beam. What is the intensity of the beam 15feet below the surface?

A classical problem in the calculus of variations is to find the shape of a curve such that a bead, under the influence of gravity, will slide from point A(0,0)to point B(x1,y1)in the least time. See Figure 3.R.3. It can be shown that a nonlinear differential for the shape y(x)of the path is y[1+(y')2]=k, where kis a constant. First solve for k dxin terms of yand dy, and then use the substitution y=ksin2θ to obtain a parametric form of the solution. The curve turns out to be a cycloid.

FIGURE3.R.3 Sliding bead in Problem 12

In the treatment of cancer of the thyroid, the radioactive liquid Iodine-131 is often used. Suppose that after one day in storage, analysis shows that an initial amount A0 of iodine-131 in a sample has decreased by 8.3% .

(a) Find the amount of iodine-131 remaining in the sample after 8 days.

(b) Explain the significance of the result in part (a) .
See all solutions

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