Homework SourseYou have a business selling widgets Your current sales level
You have a business selling widgets. Your current sales level is 10,000 widgets per month, but you expect this to decline continuously at a rate of 2 percent per month. Let S(t) denote the level of sales after t months have passed.
(a): Write down a differential equation satisfied by S(t)
S\'=???
(b): Solve the initial value problem to determine a formula for S(t).
S(t)=???
(c): After some study, you find that the above model is too inaccurate. Sales data suggests that the sales level is actually dropping 300 widgets per month MORE than initially predicted. Write down a new differential equation that incorporates this extra amount of drop in sales:
S\'=???
(d): Solve the differential equation in part C (with the initial value from the beginning of the problem) to determine the new model for future sales:
S(t)=???
(a): Write down a differential equation satisfied by S(t)
S\'=???
(b): Solve the initial value problem to determine a formula for S(t).
S(t)=???
(c): After some study, you find that the above model is too inaccurate. Sales data suggests that the sales level is actually dropping 300 widgets per month MORE than initially predicted. Write down a new differential equation that incorporates this extra amount of drop in sales:
S\'=???
(d): Solve the differential equation in part C (with the initial value from the beginning of the problem) to determine the new model for future sales:
S(t)=???
Solution
a.
S(1) = 10,000
we know that for every moth there is 2% decrease in sales
[S(1) - S(2)]/S(1) = 0.02
S(2) = S(1)[1-0.02]
similarly
[S(2) -S(3)]/S(2) = 0.02
S(3) = S(2)[1-0.02}
S(3) = S(1) [1-0.02]2
S(4) = S(1) [1-0.02]3
generalising it
S(t) = S(1) [1-0.02]t-1
S(t) = 10,000[1-0.02]t-1
S\'(t) = 10,000(t-1)[1-0.02]t-2
b.
Aolving the inital value problem we get
S(t) = 10,000[1-0.02]t-1
c.
Its identified that 300 widgets more are dropping
so there is a total 5% drop
hence
S\'(t) = 10,000(t-1)[1-0.05]t-2
c.
S(t) = 10,000[1-0.05]t-1
