This is a word problem I need help on. I would appreciate any help given! Washin

Question

This is a word problem I need help on. I would appreciate any help given!

Washington State's sentencing laws are among the most complicated in the country. Suppose there existed a function S(t) that could compute the total length of prison sentences, measured in years, imposed in the year t. Suppose the total number of convicts who were sentenced in the year t can be given by N(t). Lastly, suppose C(t) represents the average cost of imprisoning a convict sentenced in the year t, measured in dollars per convict per year. For each function, t is the number of years since 2000.

a) Suppose As(t)= S(t)/N(t) what does As(t) represent? What are the units for As(t)?

b) Find A's(t), in terms of S(t) and N(t)

c) What does A's(t) represent? What are the units of A's(t)?

d) Use following information to find A's(15). Round answers to 5 decimal places.

S(15)= 430,000 years S'(15)=-1.5 years per year N(15)=17,200 convicts N'(15)= 20 convicts per year

e) Interpret the results for A's(15)

f) Suppose Tc(t)= N(t) x C(t), what does Tc(t) represent? What are the units of Tc(t)?

Explanation / Answer

a) As(t)= S(t)/N(t)

given S(t) stands for total length of the prison sentences, in years, imposed in the year t. N(t) is th number of convicts who were sentenced in the year t.

thus from this, As(t) means the avergae length of the prison sentences imposed in the year t and its units are years per convict.

b) A's(t)=d/dt(As(t))=d/dt(S(t)/N(t))=[S(t)N'(t)-S'(t)N(t)]/N2(t)

c) A's(t)=d/dt(As(t))=d(As(t))/dt which is the change in the average length of sentence with each year and since it is a ratio of averag length in years per convict and year it happened, therefore the units of A's(t) is years per year per convict.

d) Given: S(15)=430000 years, S'(15)=-15 years per year, N(15)=17200 convicts N'(15)=20 convicts per year

A's(t)=[S(t)N'(t)-S'(t)N(t)]/N2(t) = [(430000)(20)-(-15)(17200)]/(17200)2=0.02994 years per year per convict.

e) From the results of A's(t), we com to know that the average length of prison sentence per convict is changing by 0.02994 years every year.

f) Tc(t)=N(t) x C(t)= (number of convicts who were sentenced in the year t) x (average cost of imprisoning a convict sentenced in the year t)

Thus Tc(t) gives the total cost of improsining the convicts in th year t. Since N(t) is given in number of convicts and C(t) is in dollars per convict per year, thus units of Tc(t) is dollars per year.

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