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Question 3* (a) Suppose it costs $120 plus 20 cents a mile to rent a car for a week. Let A(n) represent the total cost for renting the car if you drive for a total of n miles. Develop a dynamical system (i.e., the difference equation) for A(n+1) in terms of A(n). Solve the equation for A(n). (b) Let A(n) be the number of gallons of gas left in a car after driving n miles. The car originally had A(0) = 12 gallons, and it goes 20 miles per gallon of gas. Develop a dynamical system (i.e., the difference equation) describing the amount of gas left after driving n+1 miles in terms of the amount of gas left after driving n miles. Solve it for A(n). (c) Suppose that a person takes a pill containing 200 mg of a drug every 4 hours, and assume that the drug goes into the bloodstream immediately. Also assume that every 4 hours the body eliminates 20 percent of the drug that is in the bloodstream. Develop and solve a dynamical system describing the amount A(n) (in mg) of the drug in the bloodstream after taking the nth pill.

Answer 1

a) $A(n+1)=A(n)+20,A(0)=120$ so that $A(1)=120+20,A(2)=120+2\cdot 20\text{ and in general }A(n)=120+20n$

b) $A(0)=12,A(n+1)=A(n)-\frac{1}{20}$

Which means $A(1)=12-\frac{1}{20},A(2)=12-\frac{2}{20}\cdots A(n)=12-\frac{n}{20}$

c)

C(O) = 200, C(n + 1) = (0.8)C(n) + 200

And so $C(1)=0.8\cdot 200+200, C(2)=0.8^2\cdot 200+0.8\cdot 200+200$

And in general $C(n)=200(0.8^n+0.8^{n-1}+0.8+1)$

$C(n)=200\times \frac{1-0.8^{n+1}}{1-0.8}$

Which is $C(n)=1000\times (1-0.8^{n+1})$

$\blacksquare$

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