Suppose you are having a dinner party for a large group of people, and you decid

Question

Suppose you are having a dinner party for a large group of people, and you decide to make 2 gallons of chili. The recipe calls for 2 teaspoons of hot sauce per gallon but you misread the instructions and put in 2 tablespoons of hot sauce per gallon. (since wach tablespoon is 3 teaspoons, you have put in 6 teaspoons per gallon, which is a total of 12 teaspoons of hot sauce in the chili.) You don't want to throw it away, so you serve the chili anyway. However as each person takes so chili, you fill ip the pot with beans and tomatows without hot sauce until the concentration of hot sauce agrees with this recipe. Suppose the guests take 1 cup of chili per minute from the pot. (there is 16 cups in a gallon) how long will it take to get the chili back to the recipe's concentration of the hot sauce? How many cups of chili will have been taken from the pot?

Explanation / Answer

Where Q is quantity of teaspoons, V is volume and r is the rate of the flow, dQ/dt = -Q/V*r But it is given that V = 32 (2 gal. = 32 cups) and r = -1 (1 cup per minute is flowing out) , thus dQ/dt = -Q/32 dQ/Q = -1/32 * dt integrate both sides ln|Q| = -1/32*t + c |Q| e^(-1/32*t + c) = e^c*e^(-1/32*t) let A = e^c and drop abs val signs b/c you can't have neg. qty Q(t) = A*e^(-1/32*t) Since you know you start with 12 teaspoons at time 0 Q(0) = 12 = A*e^(-1/32*0) implies A = 12, thus Q(t) = 12*e^(-1/32*t) Now, to get the amnt you want, 4 teas, set the LHS = 4 and solve for t.

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