e e \"dl 95%. 2:29 PM K/ Please shom as mach work as you can to receive full cre

Question

e e "dl 95%. 2:29 PM K/ Please shom as mach work as you can to receive full credit. Yos may turn this sheet in or woek on a separate sheet of pape (Relevant Section: 1.2) Problem. Suppose you are having a dinner party for a large group of peopk, and yon decide to make 2 galloes of chill. The reeipe calls for 2 teaspoons of hot sasce per gallon, bat yos misead the instructi and p t in 2 tallespots of het sme per galkm. (Saeeesch tablespoon is 3 tewpoons, yolmr i26 teaspoons per gallon, which is a total of 12 tenspoons of hot sauce in the chilk) You don't want to thow the chili out bcau theee isst ch elhe to at (and sotne people li bot chilt), so you wre the chil ayway Hourm, as each pemen talos Mene e, yun fil up thr pct with braas aad tumators without Iact sauce until the concestration of hot sace agrees with the recipe. Suppose the gusts talor 1 eup of cili per ite from the pot (there are 16 cups in a galkon), how long will it take to gt tle chili back to the recipes

Explanation / Answer

if y is the number of teaspoons of hot sauce, then y(0) = 12.

dy/dt = amount of hotsauce per cup at time t (assuming t is in minutes).
= - y / 32. (Because 1/32 of the hot sauce will be in each cup).

This is a simple equation that can be solved to get that y(t) = Kexp(-t/32), where K is some constant. We know in fact that K is 12, in order to solve the initial condition.

So, we have that 12exp(-t/32) = 4.

==> exp(-t/32) = 1/3

==> -t/32 = -ln(3)

==> t = 32ln3

= 35.1555.

So, you'll need 36 cups to get below the intended concentration.

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