Question

A tank of water is contaminated with 55 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 24% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

(a) Explain why S is an exponential function.

The amount of salt is being decreased by   ---Select--- % pounds each hour, so the decay is at a  ---Select--- decreasing proportional constant proportional decreasing linear constant linear rate.

Find its hourly decay factor.


(b) Give a formula for S.
S =

(c) Make the graph of S that shows the flushing process during the first 15 hours.

Describe in words how the salt removal process progresses.

---Select--- Large Small amounts of salt are removed early in the process, and, as time passes, the rate of removal  ---Select--- increases decreases remains the same .

(d) In order to meet EPA standards, there can be no more than 3 pounds of salt in the tank. How long must the process continue before EPA standards are met? (Round your answer to two decimal places.)
hr

(e) Suppose this cleanup procedure costs $8000 per hour to operate. How much does it cost to reduce the amount of salt from 55 pounds to 3 pounds? (Round your answer to the nearest dollar.)
$

How much does it cost to reduce the amount of salt from 3 pounds to 0.1 pound? (Round your answer to the nearest whole dollar.)
$

Answer 1

Tue @ The amount of salt is being deereased by 24% each hous, so the decay is at at decreasing proportional It's howly de cay factor is 24% 1.e. 24 100 6 25 0.24 S - SCK) = 55(1-0-243€ 55.(0.76) After first 15 hours, remaining sat amount of salt is 55:(0.76)" by pounds 0.8965 pounds, M T W ] I S 1 ? 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 February 2007

e I wed so large amount of salts are removed early in the process, and, as time passes, decreases the rate rate of removal. on @ Let required time is t houss hen SLO 5 3 55.(0:76)€ < 3 (0.76€ 5.55 Choose t> log ( 5 ) Com or 3 بل - or 55 3 or In (5) In 100 76 or, t = 10.59887... (correct upto 2D places) 10. 60 must So, the process should continue 10:60 houses before EPA standards are met. 1 M T W ] f S 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3! e o reduce the salt from 95 pounds to 3 pound we must have to continue the process 10.60 hours So, the total cleaning cost will be $10.60% 8000 = $ 84 800 (approximately). Actual cost is $10.59887... X 8000 $ 84 790.97 December 2005

1 Thu Thu Now let the required time to reduce the sall from 3 pounds to oni pounds is t, hours. Then we get to reduce the amount of salt from 55 pormas to 3 ponas the required time is In ($) Iom. (House) Similarly to we get In (55) t+t, t 10.59887... ܐ lo 100 76 .. t, 2. Fri In 65 ) In 32 In 1990 in cho In (5) - ln (57) hours In / 100 20 places) Cost 12.393370040983 hours 12:39 hours (correct uph So, the the required is $ t, x overo = $ 99.146 -96 03278.665.. $ 99.147_ ( nearest Whde dollar) M T W T F S S 1 ? 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 March 2007 ~

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