Question

Suppose that a company needs new equipment, and that the machinery in question earns the company revenue at a continuous rate of 66000t 38000 dollars per year during the first six months of operation, and at the continuous rate of $71000 per year after the first six months. The cost of the machine is $160000. The interest rate is 5% per year, compounded continuously. a) Find the present value of the revenue earned by the machine during the first year of operation. Round your answer to the nearest cent. Value: $ equation editorEquation Editor b) Determine how long it will take for the machine to pay for itself; that is, how long until the present value of the revenue is equal to the cost of the machine. Round your answer to the nearest hundredth. Years: equation editorEquation Editor

Answer 1

The present value of revenue is calculated by discounting the company's two different revenue streams to their current worth using continuous compounding. The time for the machine to pay for itself is determined by equating the present value of revenue to the machine cost and solving for time.

Step-by-step explanation:

Calculating the present value of revenue earned by a company's machinery involves discounting the future revenue streams to their current worth given a continuous interest rate of 5%. To find the present value (PV) of the revenue earned during the first year of operation, we must consider two different revenue rates: $66,000t + $38,000 for the first six months, and $71,000 per year for the remaining six months. Discounting these amounts at a continuous compounding rate gives us the present value of the revenue for the first year.

To determine when the machine will pay for itself, we have to calculate the point in time when the cumulative present value of the revenue equals the cost of the machine, which is $160,000. This involves finding the time, to the nearest hundredth of a year, when the discounted revenues from the machinery match the initial investment, ensuring that the company's expenditure on equipment is justified by the earnings it generates.

Answer 2

a-The present value of revenue in the first year is $61,085.92.

b-The total time it would take to pay for its price is 2.44 years of 29.33 months.

Step-by-step explanation:

a-

Let the function of the revenue earned is given as

Here

• a and b are the limits of integral which are 0 and 1 respectively
• r is the rate of interest which is 5% or 0.05

• S(t) is the function of value which is
  

  So the present value of revenue in the first year is $61,085.92.

  b-

  The time in which the machine pays for itself is given as

  
  `PV=\int\limits^0_1 {S(t)e^(-0.05t)} \, dt+\int\limits^t_1 {S(t)e^(-0.05t)} \, dt\\PV=61085.9225+\int\limits^(t)_(1){(71000)e^(-0.05t)} \, dt`

  The present value is set equal to the value of machine which is given as

  $160,000 so the equation becomes:

  
  `PV=61085.9225+\int\limits^(t)_(0){(71000)e^(-0.05t)} \, dt\\160000=61085.9225+\int\limits^(t)_(0){(71000)e^(-0.05t)} \, dt\\\int\limits^(t)_(0){(71000)e^(-0.05t)} \, dt=160000-61085.9225\\\int\limits^(t)_(1){(71000)e^(-0.05t)} \, dt=98914.07\\\\t=-(\ln \left(0.93034\right))/(0.05)\\t=1.44496`

  So the total time it would take to pay for its price is 2.44 years of 29.33 months.