A construction company has an expenditure rate of E'(x)=e^(0.17 x) dollars per day on a particular paving job and an income rate of I'(x)=116.2 - e^(.17x) dollars per day on the same job, where x is the number of days from the start of the job. The company's profit on that job will equal total income less total expenditures. Profit will be maximized if the job ends at the optimum time, which is the point where the two curves meet.
(a) Find the optimum number of days for the job to last.
(b) Find the total income for the optimum number of days.
(c) Find the total expenditures for the optimum number of days.
(d) Find the maximum profit for the job.

a) optimum number of days for the job to last = 24 days

b) Total income for the optimum number of days = $2434.83

c) Total expenditures for the optimum number of days = $341.77

d) Maximum profit for the job = $2093.06

Step-by-step explanation:

a)  Expenditure rate ,

Income rate

The job lasts for many days where the income is more than expenditure and job ends when both are equal.

The optimum number of days is the value of x where I'(x) = E'(x)

The optimum number of days for the job to last is 24 days

b)

Integrating both sides in the equation above:

Substituting   into the equation above , The income for the optimum number of days becomes:

c)

Integrating the equation above:

Substituting into the above equation, expenditure for the optimum number of days is;

d) Maximum profit is 2434.83- 341.77=2093.06

yes

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