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A company buys a machine with a marginal rate of change in revenue, MR(t)=107-t, where time...

A company buys a machine with a marginal rate of change in revenue, MR(t)=107-t, where time t is measured in years. The company estimates that the marginal cost of maintaining and repairing the machine is given by the function MC(t)=t2-7t+100.

1) Calculate the operating time t* that maximizes total profits.

2) What are the total profits when the machine has no residual value in time t*?

3) What are the total profits when the machine has a residual value given by the formula S(t)=100-12t.

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Answer #1

1. Profit will be maximized when MR=MC. In our case, when

107-t=t2-7t+100

Solving, we get

t=7

2. MR is 107-t. Revenue would be integration of that so,

Revenue=107t-t2/2

Cost would be integration of MC, which means

Cost=t3/3-7t2/2+100t

Profit=Revenue-cost when t=7, so

Profit=107*7-49/2-343/3+343/2-700

Profit=245/3=81.67

3. When machine has residual value, that value will be added in profits. At t=7, residual value=100-12*7=16.

Profits=81.67+16=97.67

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