Radium was found to decompose 2% of its initial quantity after 20 years. Determine the half-life of theradioactive substance.a. What is the working equation for this problem?b. What is the general solution for the given radioactive substance?C. What is the boundary condition to determine the constant C.d.After 20 years, what is the remaining quantity of the radioactive substance with reference to its initialamount?e. What is the boundary condition to compute for the proportionality constant k?f. What is the particular equation for the given radioactive substance?g. What is the half-life of the radium in years?

NOTE: As per Bartleby guidelines for multipart question. We have to answer only first 3 subpart.…

Answered: Radium was found to decompose 2% of its…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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As we saw in Section 3.4, a radioactive substance decaysexponentially: The mass at time t is m(t) = m(0) ekt , where m (0)is the initial mass and k is a negative constant. Themean life of M an atom in the substance is M = -k integral 0 to infinity t e kt dtFor the radioactive carbon isotope, 14 C ?, used in radiocarbondating, the value of k is -0.000121 . Find the meanlife of a 14 C ? atom.
7b
1. Consider a pond that initially contains 10 million gal of fresh water. Stream water containing an undesirable chemical flows into the pond at the rate of 5 million gal/yr, and the mixture in the pond flows out through an overflow culvert at the same rate. The concentration y(t) of the incoming water varies periodically with time t, measure in years, according to the expression y(t) = 2 + sin 2t g/gal. %3D a. Write out the differential equation that models this scenario. Include the initial value as well. b. Solve the initial value problem from a. You may use technology to help with the integration. c. Plot this solution for t e (0,20) years. What is happening to this pond as time moves on?
The half-life of cesium-137 is approximately 30 years. If an artifact initially contains 100 g of cesium, find a formula for the amount of cesium in the artifact at any time T. Hint: Use y = Yoe -kT where yo is the initial value and k is the decay constant. -In(2) ·T y = 100e 30 y = 100ekT y = 100e307 y = 30e100T
1. Solve algebraically and approximate solutions to three decimal places. a. In(x- 3) = 0.924 %3D b. e103x =7 %3D c. log, x+log, (8-x)=2 4.5 = 4e2.1x +3.3 %3D d.
3. The figure below represents the cross-section of a tank. The length of the tank is L. (For example, the area of the bottom section of the tank is wi × L). a. When h D, give a differential equation for h. c. What is the steady-state value for h, if R = 1, w1 = 1, w2 = 2, L = 1, D = 1, p = 1, qim = 20, g = 9.8 m/s? R
14. The decay of a certain bacteria population can be modelled by P(t) = 2000e-1.2t, where t is in hours. а. What is the initial population? b. What is the half-life? What is the population after 5.179 hours? C.
This topic is Radioactive Decay in Differential Equations.
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Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min

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