Happy Valley Pond is currently populated by yellow perch. The graph at the right
ID: 3283387 • Letter: H
Question
Happy Valley Pond is currently populated by yellow perch. The graph at the right gives an outline of the pond. The pond is fed by two springs: spring A contributes 50 gallons of water per hour during the dry season and 80 gallons of water per hour dr ing the rainy season. Spring B contributes 60 gallons of water per hour during the dry season and 75 gallons of water per hour during the rainy season. During the dry season an average of 110 gallons of water per hour evaporates from the pond, and an average of 90 gallons per hour of water evaporates during the rainy season. There is a small spillover dam at one end of the pond, and any overflow will go over the dam into Bubbling Brook B 12 12102 D C. B 2 1018 12 620 5 Scale; 20 ft 10 A Happy Valley Pond c1d5bc3b 1950cc ebra Spring B has become contaminated with salt and is now 10% salt. (This means that 10% of a gallon of water from Spring B is salt.) The yellow perch will start to die if the concentration of salt in the pond rises to 1%. Assume that the salt will not evaporate but will mix thoroughly with the water in the pond. There was no salt in the pond before the contami nation of spring B. Your group has been called on by the Happy Valley Bureau of Fisheries to try to save the perch. 1. The attached table gives a series of measurements of the depth of the pond at the indicated points when the water level is exactly even with the top of the spillover dam Calculate how many gallons of water are in the pond when the water level is exactly even with the top of the spillover dam. Explain how accurate you think your measurement is and why. (A gallon is 231 cubic inches or 231/1728-0.13368 cubic feet.) 2. Let 0 hours correspond to the time when Spring B became contaminated. Assume it is the dry season and that at time0 the water level of the pond was exactly even with the top of the spillover dam. (a) How much salt will be in the pond after t hours? Calculate the amou f0, 12,24, 36. 48,60, 72 (b) What is the percentage of salt in the pond after t hours? Calculate the percentage for t0, 12,24,36,48, 60, 72 (c What will the change in the amount of salt in the pond be during the time interval to + h for some 674644bf5c 50cca2 positive h (d) Write a differential equation for the amount of salt in the pond after t hours (e) Solve the differential equation and find an expression for the amount of salt in the pond after t hours (f) Draw a graph of the amount of salt in the pond versus time for the next three months. (g) How much salt will there be in the pond in the long run? (h) Do the fish die? If so when do they start to die?Explanation / Answer
ANSWER:
After contamination by salt (lets call the amount of salt S), spring B still contributes 60 gallons/hr to the pond.
But 10% of that is now salt.
10% of 60 is 6.
The equivalent of 6 gallons/hr of salt is entering the pond. If we write this in the language of mathematics, then that will be our answer for question (a), we have:
-->Differential equation:
dS/dt=6 (units in gallons/hr)
To solve, we rewrite the equation above as dS= 6.dt then integrate:
?dS= ?6.dt
From which:
S= 6t+c (c is arbitrary constant of integration)
At t=0, the amount of salt in the pond is S=0
So, finding for c, we have c= S-6t= 0-6(0)=0
Therefore the buildup of salt in the pond over time is:
S= 6t (Units of salt equivalent to gallons, units of time in hours)
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